on 21-Nov-2023 (Tue)

In solving some ODE problems we may specify these values at the same point. These are called initial value problems(IVP)

In solving other ODE problems these values may be given at the boundaries of the domain of solution. The solution of these problems are called boundary value problems(BVP)

Based on the definition of the BVP, the definition of the two-point boundary value problems is a differential equation \(\frac{d^{2}y}{dx^{2}}+p(x)\frac{dy}{dx}+q(x)y(x)=0\) subjected to the boundary conditions \(y(a)=x_{a},y(b)=x_{b}\) where \(a\neq b\).

In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.

Pneumonia, peritonitis, urinary tract infections, and soft tissue infections are—in descending order—the most important sources of sepsis

Organ failure is the hallmark sign of sepsis and most frequently occurs in the respiratory and cardiovascular systems, followed by renal, central nervous system, hematologic, and hepatic dysfunction

A causative microorganism can be identified in up to two-thirds of patients admitted to an intensive care unit for sepsis.

Gram-negative bacteria are identified in 38% to 62% of cases, gram-positive bacteria are identified in 40% to 52%, and fungi are identified in 5% to 19%. Viral infections are detected in 1% to 7% of cases

A key pathogenic feature of sepsis is that the multifaceted innate immune response fails to return to normal homeostasis in the context of an unsuccessful attempt to eradicate the invading pathogen.

Sepsis is a broad term used for an incompletely understood process, and there is no gold standard for its diagnosis.

he term sepsis originates from the ancient Greek word sêpsis (“putrefaction” or “decay of organic matter”) and was first used in a medical context in Homer’s Iliad, written more than 2700 years ago

Sepsis-1 was defined as (documented or suspected) infection leading to the onset of SIRS as reflected by the presence of two or more SIRS criteria.

In the Sepsis-2 definition the criteria for severe sepsis remained similar, whereas septic shock was defined more explicitly as refractory hypotension (systolic blood pressure <.2

The most recent Sepsis-3 definition, published in 2016,3 seeks to confine important limitations of Sepsis-1 and Sepsis-2, which include a disproportionate emphasis on inflammation, poor specificity and sensitiv- ity of the SIRS criteria, and the incorrect concept that sepsis follows a continuum through severe sepsis to shock.

The new definition abandoned the use of SIRS criteria in the diagnosis of sepsis. In addition, in the new definition the pres- ence of organ dysfunction is a requirement for a sepsis diagnosis, and therefore the term severe sepsis was eliminated in Sepsis-3.

Septic shock is now defined as a subset of sepsis in which strong circulatory, cellular, and metabolic abnormalities are associated with a greater risk of mortality than with sepsis alone; these patients can be clinically identified by a vasopressor requirement to maintain a mean arterial pressure of 65 mm Hg or greater and serum lactate level greater than 2 mmol/L (>18 mg/dL) in the absence of hypovolemia

Importantly, qSOFA is not part of the new definition of sepsis

Moreover, failure to meet two or more SOFA or qSOFA criteria should not lead to a delay of treatment of infection or any other intervention deemed necessary by physicians.

Although large cohorts were used to support Sepsis-3 and qSOFA, patient data were almost exclusively derived from high-income countries (especially the United States), which leaves uncertainty with regard to extrapolation to resource- poor settings. In addition, supporting data were generated in adult patients and cannot be readily extrapolated to pediatric sepsis.

The manifestations of sepsis depend on the source of infection, the causative pathogen, the type and extent of organ dysfunction, drug use and comorbidity of the patient, and the delay before consulting a physician or before start of treatment.

The most common underlying comorbidities of patients admitted for sepsis include chronic obstructive pulmonary disease, neoplasm, human immunodeficiency virus (HIV) infection, chronic liver disease, chronic renal disease, diabetes, peripheral vascular disease, and autoimmune disease.6

General variables include fever, tachycardia, tachypnea, altered mental status, significant edema, or positive fluid balance (>20 mL/kg over 24 hours).2 Hypothermia is observed in 9% to 35% of patients with sepsis and is associated with adverse outcomes.7,8

Pneumonia is the most common source of sepsis in adults, followed by abdominal, urinary tract, and skin/soft tissue infections (Fig. 73.1A).6,9,10,11,12 These preferred sites account for 80% to 90% of all adult sepsis cases, the remainder being caused by bone/joint infections, ear-nose-throat infections, and others. More than one source is found in approximately 6% of episodes.9

The primary source of infection often depends on the comorbidity of the patient and the presence of indwelling catheters or devices. For instance, the presence of obstructive lung disease is a significant predictor of hospitalizations caused by respiratory tract infections but not of hospitalizations due to infections outside the respiratory system.13 Likewise, in patients on long-term renal replacement therapy, sepsis more often is caused by central venous catheter infections, peritonitis, or an ischemic bowel.14 However, in most series, no obvious source of infection can be found in one-sixth of patients.6,9,10,11,12

Organ failure is the hallmark sign of sepsis, and respiratory and cardio- vascular failure most frequently occurs, followed by renal, central nervous system (CNS), hematologic, and hepatic failure (Fig. 73.1B).5,6,10,15,16

Respiratory failure, shock, and kidney failure are the most common reasons for admission to the intensive care unit (ICU).10,15 Most patients have failure of a single organ, whereas approximately 20% have failure of two organs, and approximately 5% have acute failure of three or more organs.6,1

There is a direct relationship between the number of organs failing in patients with sepsis and mortality.10 A study from across Europe showed mortality rates were 26% among patients with dysfunction of two organs on ICU admission, whereas mortality rates were 65% among patients with failure of four or more organs.10

Neutrophils and monocytes accumulate in the lungs and may form cellular aggregates in pulmonary vessels. Significant right-to-left shunting occurs. Dead space volume increases and compliance decreases, augmenting the work of breathing. Mechanical ventilation is needed in most severe cases. In the past 2 decades, mortality among patients with ARDS has decreased from approximately 60% to 25%, which can be accounted for in part by the application of a lung- protective ventilation strategy

The cardiovascular impact of sepsis has two components: myocardial dysfunction and relative hypovolemia resulting from vasodilation.23,24

Sepsis-associated myocardial dysfunction includes reduced left and right ventricular ejection fractions, increased left and right ventricular end- diastolic volumes, and an elevated heart rate and cardiac output. The cardiac depression associated with septic shock reflects the effects of inflammatory mediators on cardiac myocyte and microcirculatory function, is not caused by ischemia, and usually does not require inotropic therapy. However, if hypovolemic shock cannot be reversed by administer- ing intravenous fluids, the use of vasopressors is indicated to restore tissue perfusion pressure

Adrenergic agonists are the first-line vasopressors, norepinephrine being the first-choice agent.25 In cases of refractory septic shock, another vasopressor with a different mechanism of action (nonadrenergic) can be added. Mechanisms implicated in the development of sepsis-induced myocardial depression include alterations in calcium homeostasis, mitochondrial dysfunction, apoptosis, circulating cardiosuppressant mediators, and nitric oxide.

Sepsis-associated AKI is associated with a high burden of morbidity (e.g., dependency on dialysis at the time of hospital discharge) and mortality in patients with critical illness.

The renal abnormalities range from minimal proteinuria to profound renal failure; postmortem studies have found focal acute tubular injury and minimal glomerular damage.26 The underlying pathogenic mechanisms include hypovolemia, hypotension, renal vasoconstriction, and toxic drugs (e.g., aminoglycosides) or contrast agents used for medical imaging.27

Brain dysfunction in sepsis may manifest as coma or delirium. Formally, sepsis-associated encephalopathy (SAE) is defined as diffuse cerebral dysfunction that accompanies sepsis in the absence of direct CNS infection, structural abnormalities, or other types of encephalopathy.31

Delirium occurs in 30% to 50% of patients with severe sepsis. In general, the severity of SAE parallels the severity of other manifestations of sepsis.

Because there are no specific markers for SAE, the diagnosis relies on excluding primary CNS infections and other causes of encephalopathy. Proposed underlying mechanisms include microscopic brain injury, blood-brain barrier and cerebral microcirculation dysfunc- tion, altered CNS metabolism, and impaired cholinergic neurotransmis- sion.

Critical illness polyneuropathy (CIP) and critical illness myopathy (CIM) are other common complica- tions in patients with prolonged ICU stay. Clinical features include difficulty in weaning from a ventilator, generalized wasting of the limbs, and diffuse weakness.33 The diagnosis of CIP and CIM relies on clinical, electrophysiologic, and muscle biopsy investigations.33 The septic inflammatory response is thought to play an important role in their pathogenesis.34 CIP and CIM are associated with prolonged ICU stays and increased mortality.33,34

In patients with sepsis the reported prevalence of the most severe form of coagulopathy, dissemi- nated intravascular coagulation (DIC), is 35%.35

Commonly used screening assays for DIC include (1) a reduced or downward trend in the platelet count (usually <00,000/mm3); (2) the presence of fibrin- related markers including fibrin degradation products, D-dimers, or soluble fibrin in plasma; (3) prolongation of PT or APTT (>1.2 times the upper limit of normal); and (4) low plasma levels of endogenous anticoagulants such as antithrombin and protein C.36

Clinically, severe DIC can be characterized by widespread thrombosis in small and midsize vessels with simultaneous hemorrhage from various sites (Fig. 73.2).

Earlier studies have shown that the development of DIC in patients with sepsis can double the risk of death.35 At the present time, no specific therapy for DIC exists apart from treatment of the underlying disorder.25

Recombinant activated protein C, which was originally recommended in the 2004 and 2008 Surviving Sepsis Campaign guidelines, was not shown to be effective in adult patients with septic shock in the PROWESS-SHOCK trial and was subsequently withdrawn from the market.25,40

Gastrointestinal tract injuries include disruption of the intact intestinal epithelium, which may lead to the translocation of inflammatory mediators; the occurrence of erosions of the gastric and duodenal mucosa that predispose to upper gastrointestinal bleeding; and the development of ileus, which may persist for several days after the resolution of septic shock (Fig. 73.2).41,42

Hepatic dysfunction includes cholestatic jaundice characterized by elevations in conjugated and unconjugated bilirubin, often seen in association with elevated levels of alkaline phosphatase and amino- transferase levels. Preexisting liver disease can aggravate these values

A “shock liver” is unusual, but if the duration of septic shock is prolonged, a massive rise in serum transaminases may follow hypoxic necrosis of centrilobular liver cells

Other common manifestations of sepsis include altered glycemic control, adrenal dysfunction, and sick euthyroid syndrome.5

In high-income countries, sepsis represents approximately 6% of adult hospitalizations and 10% to 37% of ICU admissions.6,10,49 The overall incidence rate of hospitalization among emergency medical services encounters is greater for sepsis than for acute myocardial infarction or stroke.50

A retrospective cohort study of 173,690 adult patients with sepsis admitted to hospitals across the United States in 2014 showed that 54.7% required ICU care during hospitalization, and 15.8% had septic shock.49 Median ICU length of stay was 5 days (range, 2–6 days).49 Median hospital length of stay was 10 days (range, 8–12 days).49 The cost of treating sepsis in US hospitals was estimated to be $24 billion in 2013, making it the most expensive condition treated in US hospitals in that year.44

In the United States, sepsis contributes to one in every two to three in-hospital deaths51 and represents the most frequent cause of death in noncoronary ICUs.46

The case-fatality rate depends on the setting and severity of disease. US data from 2014 show that of all patients admitted for sepsis, 15.0% died in the hospital, and 6.2% were discharged to a hospice.49 Data from Australia and New Zealand show hospital fatality rates for sepsis and septic shock of 14% and 22%, respectively.52

P. aeruginosa can cause various specific clinical skin, soft tissue, and bone infections. Certain syndromes are classically or most commonly associated with P. aeruginosa, specifically. These include:

● Ecthyma gangrenosum

● Burn wound infections

● Folliculitis associated with hot tub exposures

● Foot infection following nail puncture injury

● Green nail syndrome

● Perichondritis following ear piercing or acupuncture

Additionally, P. aeruginosa can be involved in other general skin and soft tissue infections, such as cellulitis (particularly in neutropenic patients), postoperative infections, infections following trauma (particularly after injury in aquatic environments), and infections of chronic decubitus ulcers.

Inflammatory subcutaneous nodules (ie, panniculitis) due to P. aeruginosa in the absence of bacteremia or classic immunosuppression have been rarely reported [1].

As with other infections, P. aeruginosa soft tissue infections are often associated with worse outcomes than other pathogens. As an example, in a prospective study of 132 patients who received skin grafts for soft-tissue defects, P. aeruginosa was the most common etiologic microorganism in skin-graft loss due to infection (58 percent) [2]. Infections due to P. aeruginosa were more fulminant and were 4.2 times more likely to require reoperation that those caused by other organisms

In general, treatment of P. aeruginosa skin, soft tissue, and bone infections should include aggressive surgical debridement of any necrotic tissue and infected eschars, in addition to antibiotic therapy. Antibiotic options include beta-lactams, carbapenems, or fluoroquinolones that have antipseudomonal activity (table 1). In general, aminoglycosides should not be used as a single agent for infection at these sites.

A recent study assessing the susceptibility of P. aeruginosa strains causing SSTIs that were isolated from 47 centers from various countries reported that 98.6 percent of strains were susceptible to ceftolozane-tazobactam, 98.3 percent were susceptible to ceftazidime-avibactam, and 98.3 percent were susceptible to imipenem-relebactam [6]

P. aeruginosa has been classically associated with a characteristic skin lesion called ecthyma gangrenosum (picture 1), which is most frequently described in the setting of bacteremia in immunocompromised patients. It results from perivascular bacterial invasion of the media and adventitia of arteries and veins with secondary ischemic necrosis.

Ecthyma refers to an ulcerative lesion that extends through the epidermis and deep into the dermis and appears as a "punched-out" ulcer covered in crust surrounded by raised violaceous margins. Although ecthyma gangrenosum is not pathognomonic of P. aeruginosa infection, the presence of these lesions should immediately raise the high probability that P. aeruginosa is the causative organism. This is particularly important for critically ill patients who present with severe sepsis and require prompt antipseudomonal empiric antimicrobial therapy [12,13]

Ecthyma gangrenosum is classically associated with P. aeruginosa sepsis and bacteremia in immunocompromised patients, including those with neutropenia or congenital immunodeficiencies (eg, Bruton agammaglobulinemia [14,15]) and those on immune modulating agents. It has been observed in approximately 1.3 to 3 percent of P. aeruginosa bacteremia cases [16].

In some cases, ecthyma has been described in patients who were subsequently discovered to have an immune deficit [ 17]. Furthermore, it has also been reported in healthy, immunocompetent patients without bacteremia or systemic infection. In such patients, ecthyma gangrenosum can present as localized skin lesions [16].

Lesions may occur after breakdown of mechanical defense barriers and local infection [18]. Diabetes and malnutrition have been identified as predisposing factors in patients without classic immunocompromise [18].

The lesions of ecthyma gangrenosum (picture 1 and picture 2) commonly begin as painless red macules which rapidly evolve into areas of induration that develop into pustules and/or bullae. Ultimately, these become gangrenous ulcers. Ecthyma lesions typically progress rapidly (within 12 to 18 hours). The lesions may involve the skin or mucous membranes.

Although ecthyma gangrenosum can occur at any anatomic location, the anogenital and axillary areas are most commonly involved [16]. The gluteal and perineal areas are involved in 57 percent of cases, extremities in 30 percent, the trunk in 6 percent, and the face in 6 percent. Periorbital lesions are very rare but have been described.

Lesions may be single or multiple; in the latter case, they may be in different stages of development at any given point in time.

Biopsy of ecthyma gangrenosum lesions typically shows hemorrhagic necrosis, gram-negative bacilli, and damage to the media of arterial blood vessels [19].

Most patients have fever and may be systemically ill; however, ecthyma gangrenosum can occur in patients in the absence of fever or other constitutional signs.

Among bacterial pathogens, Pseudomonas stutzeri, Aeromonas spp, Stenotrophomonas spp, Citrobacter spp, methicillin-resistant S. aureus, and atypical mycobacteria have been described to cause ecthyma gangrenosum. Among fungal infections, Fusarium spp have been classically associated with ecthyma gangrenosum, and a few reports have described ecthyma gangrenosum-like lesions in immunocompromised patients with disseminated candidiasis [21,22].

Ecthyma gangrenosum is not pathognomonic of P. aeruginosa infection. Identical lesions have been seen with systemic infection due to an array of common and uncommon bacterial, viral, and fungal pathogens [20].

A specific microbial diagnosis is important given that many different organisms can cause ecthyma gangrenosum, which is another reason for collecting blood cultures. Culture of exudates from an aspirate or swab of lesions is easy to perform and may identify a causative pathogen. If both blood cultures and cultures from skin exudates are negative, a biopsy of the lesion for pathology and bacterial, fungal, and mycobacterial cultures may be useful to exclude other etiologies. In fact, an early skin biopsy may be beneficial in immunocompromised hosts to rule out nonbacterial causative organisms like fungi that can be visualized on direct observation of specimens and require prompt antifungal treatment [23].

Although not generally warranted for infections with limited involvement, surgical debridement to achieve source control and even skin grafting may be necessary in patients with severe or extensive skin [25].

Chronic obstructive pulmonary disease (COPD) is characterized by airway inflammation and parenchymal destruction.

Airflow obstruction produced by structural changes in the lung predisposes to acute declines in patient well-being associated with increasing dyspnea and sputum production; these are the essence of the diagnosis of acute exacerbation of chronic obstructive pulmonary disease (AECOPD).

By 2030, 10% of the world’s population will be expected to develop COPD

COPD, including emphysema and chronic bronchitis, has been the third leading cause of death since 2008 in the United States.1

Smoking tobacco is the largest risk factor; 50% of smokers will develop COPD.5–8

COPD affects predominately older individuals, with study cohorts having mean ages older than 60 years; however, presentation in young adulthood occurs in susceptible individuals.12,13,14

Tobacco smoking is the largest environmental risk factor in the United States. In other countries, occupational dust exposures, outdoor air pollution, and poor indoor air quality from burning biomass fuels also are major COPD risk factors.18

Among immunocompetent adults, increased airway bacteria produce airway inflammation and accelerate airway obstruction.19 Immunosuppression is an independent risk factor for COPD. Immunoglobulin A (IgA)- deficient individuals have repeated lower respiratory tract infections during childhood and poor adult lung function.20

Homozygous α1-antitrypsin deficiency (PI*ZZ) occurs in 1% to 4.5% of COPD patients, and the heterozygous form (PI*MZ), with less severe deficiency, occurs in 17.8% of COPD patients.26

Poor socioeconomic status, chronic asthma, intrauterine growth retarda- tion, poor nourishment, and history of pulmonary tuberculosis are other risk factors for COPD.27,30–32

Frequent exacerbators may be defined as COPD patients who have two or more exacerbations per year and likely constitute a distinct subgroup within COPD with poor outcome.33,34

Risk factors include viral and bacterial infections, change in environmental conditions such as smog, gastro- esophageal reflux, lack of compliance with maintenance treatment, and severity of baseline disease.33 Important to note, the best predictor for exacerbations is a past history of exacerbations.35

AECOPD occurs approximately two times more frequently in winter than in summer, both in northern and southern latitudes.

AECOPD manifests with progressive shortness of breath, cough, sputum production, reduced energy, and exercise limitation. These symptoms are associated with accelerated decline in lung function, which may continue despite smoking cessation

The dyspnea usually starts during exercise but can occur with minimal exertion or at rest as disease progresses.

Cough and sputum production are usually intermittent and more pronounced in the morning.

Other chronic pulmonary diseases with similar clinical presentation and acute exacerbations should be differentiated from COPD because treatment differs. Examples of these include asthma, cystic fibrosis, bronchiectasis, diffuse panbronchiolitis, and obliterative bronchiolitis.

Worsening symptoms, increased sputum volume, and transition of sputum color from clear to green or yellow suggest AECOPD, which more commonly occurs during winter months.36

The main differential diagnosis of exacerbations in patients with COPD includes pneumonia and congestive heart failure, both of which are common comorbidities in these patients.5,8

COPD is characterized by progressive airflow obstruction defined by reduction in forced expiratory volume in 1 second (FEV1) and a postbronchodilator ratio of FEV1/forced vital capacity (FVC) less than 70% on pulmonary function tests (PFTs).

A highly affected subgroup of patients have characteristics of both COPD and asthma (asthma-COPD overlap) with partial bronchodilator reversibility of airflow obstruction.37

Although there is no universal agreement on how to define or diagnose AECOPD, it is commonly defined as an acute event with worsening respiratory symptoms beyond normal day-to-day variation. The definition often requires increased rescue β-agonist inhaler use or addition of a long-acting muscarinic antagonist (LAMA) to control symptoms.38,39

The differential diagnosis must include acute coronary syndrome, conges- tive heart failure, pulmonary embolism, and pneumonia.8

One widely used scale to diagnose the presence and severity of AECOPD requires patients to have at least one of the following clinical presentations: upper respiratory infection symptoms within the prior 7 days, increased wheezing, fever without another identified cause, or an increase in heart rate or respiratory rate >20% from baseline. This scale then categorizes patients into three groups based on whether they have worsening dyspnea, increase in sputum purulence, increase in sputum volume, or a combination of these. A severe exacerbation meets all three criteria, a moderate exacerbation meets two, and a mild exacerbation meets only one criterion.40

Although clinical presentation is the most important component in establishing the diagnosis of AECOPD, procalcitonin is a biomarker that may be useful in determining the need for antibiotics.41 No biomarkers of systemic inflammation reproducibly predict future exacerbations.42

Risk factors associated with COPD exacerbations include having two or more COPD exacerbations in the previous year, reduced FEV1, smoking, and nonadherence with O2 therapy.43,44

Tachypnea (especially with a respira- tory rate above 25), tachycardia, inability to speak full sentences, and fatigue are indications for hospitalization.

Oxygen saturation above 90% can be misleading because hypoxemia is frequently a late event in the progression to respiratory failure. In an acutely symptomatic patient, arterial blood gas measurement is more useful than oximetry because only the former can enable diagnosis of hypoventilation with hypercapnia.

Use of accessory muscles with paradoxical breathing characterized by inward motion of the abdomen during inspiration indicates diaphragmatic fatigue and impending respiratory failure.

Abdominal paradoxical breathing, progressive hypercapnia, or deteriorat- ing mental status usually indicates the need for ventilatory support in an intensive care unit (ICU) with noninvasive or invasive positive pressure ventilation.5,8

Additional diagnostic tests include chest radiography to identify pulmonary infiltrates and electrocardiography to assess for cardiac ischemia and arrhythmias, particularly paroxysmal atrial tachycardia

Pulmonary emboli can mimic AECOPD, and if they are suspected, a helical CT scan with contrast can reveal this alternate diagnosis

Radiology Although chest radiography is an insensitive test to diagnose AECOPD, it is the usual first step in the evaluation of patients with progressive dyspnea and cough. It is essential for evaluation of alternative diagnoses such as pneumonia, congestive heart failure, lung cancer, or pulmonary fibrosis

Impaired lung function in COPD is caused by destruction and remodeling of large and small airways due to chronic inflammation caused by complex interactions among the inhalants (dust or smoke), the lung microbiome, and the immune response in airway mucosa.

Early pathologic condition is produced by inflammation in bronchioles less than 2 mm in diameter followed by parenchymal remodeling.46,47 In early stages, the central airway walls are infiltrated with CD8+ lymphocytes producing bronchial wall thicken- ing evidenced on chest CT.48

Pulmonary hypertension due to loss of the pulmonary capillary bed can also develop in COPD. Chronic hypoxia produces vasoconstriction, which leads to fixed structural changes that worsen pulmonary hypertension.55

As disease progresses, frequent exacerbations of COPD further contribute to increased lung inflammation and persistent loss of lung func- tion.56,57,58–60 The biology of exacerbations in COPD is heterogeneous and involves varying degrees of inflammation and microbial pathogenesis.61