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## Abstract

Background. The Chronic Hepatitis Cohort Study (CHeCS), a dynamic prospective, longitudinal, observational cohort study, was created to assess the clinical impact of chronic viral hepatitis in the United States. This report describes the cohort selection process, baseline demographics, and insurance, biopsy, hospitalization, and mortality rates.

Methods. Electronic health records of >1.6 million adult patients seen from January 2006 through December 2010 at 4 integrated healthcare systems in Detroit, Michigan; Danville, Pennsylvania; Portland, Oregon; and Honolulu, Hawaii were collected and analyzed.

Results. Of 2202 patients with chronic hepatitis B virus (HBV) infection, 50% were aged 44–63 years, 57% male, 58% Asian/Pacific Islander, and 13% black; and 5.1% had Medicaid, 16.5% Medicare, and 76.3% private insurance. During 2001–2010, 22.3% had a liver biopsy and 37.9% were hospitalized. For the 8810 patients with chronic hepatitis C virus (HCV) infection, 75% were aged 44–63 years, 60% male, 23% black; and 12% had Medicaid, 23% Medicare, and 62% private insurance. During 2001–2010, 38.4% had a liver biopsy and 44.3% were hospitalized. Among persons in care, 9% of persons with HBV and 14% of persons with HCV infection, mainly those born during 1945–1964, died during the 2006–2010 five-year period.

Conclusions. Baseline demographic, hospitalization, and mortality data from CHeCS highlight the substantial US health burden from chronic viral hepatitis, particularly among persons born during 1945–1964.

In the United States, an estimated 3.2 million persons have chronic hepatitis C virus (HCV) infection and another 1.0 million have chronic hepatitis B virus (HBV) infection [1–3]. However, chronic viral hepatitis is not widely recognized as a serious public health problem. As a result, viral hepatitis prevention, control, and surveillance programs have not been adequately resourced [4]; it has been dubbed the “silent epidemic” by Dr Howard Koh, the current US Assistant Secretary of Health [5]. To assess the larger implications and impact of chronic hepatitis B and C on the US population, we designed the Chronic Hepatitis Cohort Study (CHeCS).

The study is being carried out during a period of rapid development of viral hepatitis treatment alternatives and evolution in the healthcare system, and there are many important reasons for such an effort. For purposes of treatment and care, clinicians and others need current information on the spectrum of disease and characteristics of persons in care; types of therapies in use and their evolving benefits, risks, and adverse effects; and other factors influencing outcomes. From the larger public health and policy perspective, there is an ongoing need for information about the health burden and costs of care and mortality, implementation and effectiveness of recommended screening and care practices, access and barriers to care and treatment (who is in care and who is receiving therapy), and the costs and potential savings of appropriate care and treatment. Such information is not available from controlled clinical trials or interventional studies that, while essential for determining optimal therapy for hepatitis B and C, are limited by expense, short duration, and small numbers.

CHeCS, a “dynamic” multicenter cohort study—with rolling admissions and follow-up from patients’ regular visits to their clinicians—was designed along principles used for a large Centers for Disease Control and Prevention (CDC)–funded study of HIV infection, the HIV Outpatient Study [6]. The purpose of this analysis is to describe the cohort selection process and baseline patient population characteristics.

## METHODS

The initial cohort was created based on electronic medical and billing electronic health records (EHRs) of patients aged ≥18 years who had a service provided during 2006–2008 at 1 of 4 sites: Geisinger Health System, Danville, Pennsylvania; Henry Ford Health System, Detroit, Michigan (coordinating center); Kaiser Permanente Northwest, Portland, Oregon; and Kaiser Permanente, Honolulu, Hawaii. EHRs and billing data were collected for each patient from their first system encounter through 2010 and supplemented with individual chart review, and included patient demographics, medical encounters, laboratory results, diagnoses, and procedures including liver biopsy. Electronic data used in this analysis were available retrospectively through 1997 at the Detroit and Portland sites, through 1998 from the Hawaii site, and through 2001 from the Danville site. Analyses for the tables were limited to the years (2001–2010) for which complete data were currently available. Household income data were estimated from census tract data using geocoded addresses.

The study protocol was reviewed by an institutional review board approved by the federal Office for Human Research Protections at each participating site.

### Cohort Selection

Algorithms for inclusion in the chronic hepatitis B and C cohorts were developed and applied to the EHR and administrative data of patients aged ≥18 years from all sites with any healthcare utilization during 2006–2008. The complex algorithms were developed with a bias to capture the greatest number of verifiable chronic hepatitis B and C cases from the raw observational data, while excluding those with a single unconfirmed diagnosis or laboratory evidence that might be due to acute disease, error, or lack of necessary clinical or laboratory workup. Complete observation time for each patient was determined to be time from first evidence of hepatitis infection in the EHR, including retrospective data prior to 2006, until last health system encounter, death, or 31 December 2010.

Electronic data from 2006–2008 were reviewed to determine enrollment candidacy, and data from candidate patients were reviewed from their earliest health system encounter through 2008 to determine cohort eligibility. Patients were included in the chronic hepatitis B and C cohorts based on fulfillment of a combination of laboratory-based and International Classification of Diseases, Ninth Revision (ICD-9)–based criteria (Figures 1 and 2). In brief, for inclusion in the hepatitis B cohort, patients had to fulfill at least 2 criteria (ie, 2 positive laboratory tests consistent with current HBV infection [positive for HBV surface antigen, e-antigen, or DNA test]), or a positive laboratory test and an ICD-9 diagnosis code, or 2 ICD-9 diagnosis codes obtained at least 6 months apart (Figure 1). Patients in all phases of chronic HBV infection (immune-tolerant, immune-active, and inactive or “immune carrier”) were included. For the hepatitis C cohort, a similar approach was used: patients were included who had a positive hepatitis C RNA test, or ICD-9 diagnosis codes indicative of hepatitis C infection (Figure 2) at least 6 months apart from each other or from a positive hepatitis C antibody test or alanine aminotransferase elevation.

Figure 1.

Hepatitis B cohort inclusion criteria. 1Qualifying chronic hepatitis B International Classification of Diseases, Ninth Revision (ICD-9) codes: 070.22, 070.23, 070.32, 070.33; acute/unspecified hepatitis B ICD-9 codes: 070.20, 070.21, 070.30, 070.31. 2Qualifying lab tests: hepatitis B surface antigen (HBsAg), hepatitis B e-antigen (HBeAg), hepatitis B virus (HBV) DNA. 3Laboratory-based inclusion categories: category 1 (n = 1853), a viral hepatitis B or chronic liver disease ICD-9 diagnosis (571.5 cirrhosis of liver without mention of alcohol, 456.0–456.1 esophageal varices, 789.59 other ascites, 155.0 liver cancer, V42.7 liver replaced by transplant, V49.83 awaiting organ transplant status) plus positive laboratory evidence of HBsAg or HBV DNA; category 2 (n = 1188), ≥2 positive laboratory results for HBsAg, HBeAg, or HBV DNA, occurring at least 6 months apart; category 3 (n = 529), a negative hepatitis B immunoglobulin M core antibody (IgM anti-HBc) lab result concurrent or prior to a positive HBsAg or HBV DNA; category 4 (n = 1180), a positive hepatitis B core antibody (anti-HBc) and positive HBsAg; category 5 (n = 838), a positive HBsAg and an elevated alanine aminotransferase occurring at least 6 months apart. 4ICD-9 inclusion criterion: ≥2 ICD-9 diagnoses indicative of viral hepatitis B1 at least 6 months apart, in the absence of a positive HBsAg or HBV DNA. Abbreviations: HBV, hepatitis B virus; ICD-9, International Classification of Diseases, Ninth Revision.

Figure 1.

Hepatitis B cohort inclusion criteria. 1Qualifying chronic hepatitis B International Classification of Diseases, Ninth Revision (ICD-9) codes: 070.22, 070.23, 070.32, 070.33; acute/unspecified hepatitis B ICD-9 codes: 070.20, 070.21, 070.30, 070.31. 2Qualifying lab tests: hepatitis B surface antigen (HBsAg), hepatitis B e-antigen (HBeAg), hepatitis B virus (HBV) DNA. 3Laboratory-based inclusion categories: category 1 (n = 1853), a viral hepatitis B or chronic liver disease ICD-9 diagnosis (571.5 cirrhosis of liver without mention of alcohol, 456.0–456.1 esophageal varices, 789.59 other ascites, 155.0 liver cancer, V42.7 liver replaced by transplant, V49.83 awaiting organ transplant status) plus positive laboratory evidence of HBsAg or HBV DNA; category 2 (n = 1188), ≥2 positive laboratory results for HBsAg, HBeAg, or HBV DNA, occurring at least 6 months apart; category 3 (n = 529), a negative hepatitis B immunoglobulin M core antibody (IgM anti-HBc) lab result concurrent or prior to a positive HBsAg or HBV DNA; category 4 (n = 1180), a positive hepatitis B core antibody (anti-HBc) and positive HBsAg; category 5 (n = 838), a positive HBsAg and an elevated alanine aminotransferase occurring at least 6 months apart. 4ICD-9 inclusion criterion: ≥2 ICD-9 diagnoses indicative of viral hepatitis B1 at least 6 months apart, in the absence of a positive HBsAg or HBV DNA. Abbreviations: HBV, hepatitis B virus; ICD-9, International Classification of Diseases, Ninth Revision.

Figure 2.

Hepatitis C cohort inclusion criteria. 1Qualifying chronic hepatitis C International Classification of Diseases, Ninth Revision (ICD-9) diagnosis codes: 070.44, 070.54, 070.70, 070.71; acute/unspecified hepatitis C ICD-9 codes: 070.41, 070.51. 2Qualifying lab tests were a positive or detectable hepatitis C virus antibody (anti-HCV) test, and/or HCV RNA test, and/or hepatitis C genotype test. 3ICD-9–based categories: category 1 (n = 3506): ≥2 ICD-9 diagnoses indicating viral hepatitis C1, separated by at least 6 months at any time during the patient's medical history; category 2 (n = 2877), a viral hepatitis C1 or qualifying chronic liver disease ICD-9 diagnosis (571.5 cirrhosis of liver without mention of alcohol, 456.0–456.1 esophageal varices, 789.59 other ascites, 155.0 liver cancer, V42.7 liver replaced by transplant, V49.83 awaiting organ transplant status) separated at least 6 months apart from a positive anti-HCV test. Abbreviations: HCV, hepatitis C virus; ICD-9, International Classification of Diseases, Ninth Revision.

Figure 2.

Hepatitis C cohort inclusion criteria. 1Qualifying chronic hepatitis C International Classification of Diseases, Ninth Revision (ICD-9) diagnosis codes: 070.44, 070.54, 070.70, 070.71; acute/unspecified hepatitis C ICD-9 codes: 070.41, 070.51. 2Qualifying lab tests were a positive or detectable hepatitis C virus antibody (anti-HCV) test, and/or HCV RNA test, and/or hepatitis C genotype test. 3ICD-9–based categories: category 1 (n = 3506): ≥2 ICD-9 diagnoses indicating viral hepatitis C1, separated by at least 6 months at any time during the patient's medical history; category 2 (n = 2877), a viral hepatitis C1 or qualifying chronic liver disease ICD-9 diagnosis (571.5 cirrhosis of liver without mention of alcohol, 456.0–456.1 esophageal varices, 789.59 other ascites, 155.0 liver cancer, V42.7 liver replaced by transplant, V49.83 awaiting organ transplant status) separated at least 6 months apart from a positive anti-HCV test. Abbreviations: HCV, hepatitis C virus; ICD-9, International Classification of Diseases, Ninth Revision.

Trained medical abstractors reviewed the EHRs of patients identified as eligible from first system encounter through 2010 to collect liver biopsy results, outside system laboratory results, and detailed antiviral therapy data during 2006–2010 as well as summary history of antiviral therapy prior to 2006, and to confirm chronic infection status. Abstractors flagged charts lacking documentation that a liver specialist had diagnosed chronic hepatitis B or C, or with documentation that the patient had been diagnosed with acute hepatitis or had chronic hepatitis ruled out. Flagged charts were reviewed under the supervision of a hepatitis clinician using hepatologist-developed criteria. Cases for which chronic infection had been ruled out were excluded from the cohort. Some cases were determined to have insufficient follow-up to confirm chronic hepatitis infection status. To assess the validity of our electronic inclusion criteria, we analyzed exclusion rates among the ICD-9–based inclusion criteria and laboratory-based criteria.

### Mortality

Deaths from all causes among cohort patients that occurred or were reported to the health system facilities during 2006–2010 were enumerated. Each health system compared records to the Social Security Death Index or National Death Index to enhance ascertainment. As patients were selected for cohort eligibility based on health system service encounters during 2006–2008, the start of observation for death rate calculation was the later of 1 January 2006 or first hepatitis diagnosis date. Observation time was truncated at 31 December 2010 or date of death, whichever was earlier. All-cause death rates were calculated and for HCV compared with published National Health and Nutrition Examination Survey (NHANES) III data on patients with and without chronic HCV infection [7].

## RESULTS

### Chronic Hepatitis B Cohort

We examined the records of 1 652 055 patients aged ≥18 years in the 4 participating health systems who had 1 or more services provided during 2006–2008, and a total of 2535 met the electronic hepatitis B cohort inclusion criteria (Figure 1). After exclusion of patients for whom chronic infection was not confirmed by abstractor records review, 2202 (87%) remained in the cohort (Figure 1). Median observation time was 5.3 years (range, <1 to 16 years), with a total of 14 034 person-years of observation.

Demographic and clinical features of the 2202 confirmed chronic hepatitis B cohort patients are shown in Table 1. About half (47.9%) were born between 1945 and 1964, and an additional 22.8% were born between 1965 and 1974. Racial mix varied by study site, ranging from 21.9% Asian/Pacific Islander in Danville to 95.0% in Hawaii, and from 1% black in Hawaii to 36% in Detroit. Household income (estimated from census tract geocode) was less than $30 000/year for 13.6% of patients overall, ranging from 7.7% in Portland to 31.2% in Danville. Overall insurance coverage included 5.1% with Medicaid, 16.5% Medicare, and 76.3% private insurance, but payer mix varied by site. The percentage of patients having only public insurance (Medicaid or Medicare) ranged from 1.7% in Portland to 27.5% in Danville; 11.4% of patients in Danville and 4.0% in Detroit were uninsured. Table 1. Demographic, Clinical, and Laboratory Characteristics of Patients With Chronic Hepatitis B Characteristic Total (N = 2202) Portland, Oregon (n = 776; 35.2%) Hawaii (n = 653; 29.7%) Detroit, Michigan (n = 606; 27.5%) Danville, Pennsylvania (n = 167; 7.6%) Demographic Characteristic Birth year cohort (age in 2008) 1985–1990 (18–23 y) 54 (2.5%) 19 (2.4%) 20 (3.1%) 10 (1.7%) 5 (3.0%) 1975–1984 (24–33 y) 338 (15.3%) 128 (16.5%) 114 (17.5%) 62 (10.2%) 34 (20.4%) 1965–1974 (34–43 y) 501 (22.8%) 192 (24.7%) 130 (19.9%) 147 (24.3%) 32 (19.2%) 1955–1964 (44–53 y) 602 (27.3%) 215 (27.7%) 159 (24.3%) 183 (30.2%) 45 (26.9%) 1945–1954 (54–63 y) 452 (20.5%) 148 (19.1%) 132 (20.2%) 139 (22.9%) 33 (19.8%) 1935–1944 (64–73 y) 177 (8.0%) 55 (7.1%) 68 (10.4%) 47 (7.8%) 7 (4.2%) ≤1934 (≥74 y) 78 (3.5%) 19 (2.4%) 30 (4.6%) 18 (3.0%) 11 (6.6%) Sex Female 954 (43.3%) 338 (43.6%) 331 (50.7%) 223 (36.8%) 62 (37.1%) Male 1248 (56.7%) 438 (56.4%) 322 (49.3%) 383 (63.2%) 105 (62.9%) Race/ethnicity Race status known 1803 (81.9%) 584 (75.3%) 518 (79.3%) 546 (90.1%) 155 (92.8%) White 510 (28.3%) 198 (33.9%) 18 (3.5%) 187 (34.2%) 107 (69.0%) Black 239 (13.3%) 23 (3.9%) 5 (1.0%) 197 (36.1%) 14 (9.0%) Asian 843 (46.8%) 345 (59.1%) 309 (59.7%) 155 (28.4%) 34 (21.9%) Pacific Islander 198 (11.0%) 15 (2.6%) 183 (35.3%) American Indian/Alaska Native 13 (0.7%) 3 (0.5%) 3 (0.6%) 7 (1.3%) Ethnicity status known 1706 (77.5%) 493 (63.5%) 480 (73.5%) 566 (93.4%) 167 (100.0%) Hispanic ethnicity 39 (2.3%) 10 (2.0%) 11 (2.3%) 6 (1.1%) 12 (7.2%) Estimated household income (census tract geocode) Status known 2132 (96.8%) 767 (98.8%) 633 (96.9%) 575 (94.9%) 157 (94.0%) <$15 000 33 (1.5%) 4 (0.5%) 4 (0.6%) 17 (3.0%) 8 (5.1%)
$15 000–$29 000 258 (12.1%) 55 (7.2%) 63 (10.0%) 99 (17.2%) 41 (26.1%)
$30 000–$49 000 904 (42.4%) 370 (48.2%) 263 (41.5%) 180 (31.3%) 91 (58.0%)
$50 000–$75 000 674 (31.6%) 260 (33.9%) 248 (39.2%) 150 (26.1%) 16 (10.2%)
>$75 000 263 (12.3%) 78 (10.2%) 55 (8.7%) 129 (22.4%) 1 (0.6%) Insurance Status known 2096 (95.2%) 698 (89.9%) 625 (95.7%) 606 (100.0%) 167 (100.0%) Medicaid 107 (5.1%) 12 (1.7%) 36 (5.8%) 36 (5.9%) 23 (13.8%) Medicare only 23 (1.1%) 23 (13.8%) Medicare Plus 323 (15.4%) 70 (10.0%) 90 (14.4%) 163 (26.9%) Private 1600 (76.3%) 616 (88.3%) 499 (79.8%) 383 (63.2%) 102 (61.1%) None 43 (2.1%) 24 (4.0%) 19 (11.4%) Clinical or Laboratory Characteristic ALT Ever measured during 2001-2010a 2173 (98.7%) 768 (99.0%) 644 (98.6%) 595 (98.2%) 166 (99.4%) Highest ever ALT in 2001–2010a Less than ULN 1103 (50.8%) 401 (52.2%) 397 (61.6%) 237 (39.8%) 68 (41.0%) >ULN to ≤2 × ULN 457 (21.0%) 171 (22.3%) 123 (19.1%) 128 (21.5%) 35 (21.1%) >2 × ULN 613 (28.2%) 196 (25.5%) 124 (19.3%) 230 (38.7%) 63 (38.0%) Most recent ALT in 2001–2010a Less than ULN 1802 (82.9%) 624 (81.3%) 572 (88.8%) 485 (81.5%) 121 (72.9%) >ULN to ≤2 × ULN 253 (11.6%) 103 (13.4%) 53 (8.2%) 68 (11.4%) 29 (17.5%) >2 × ULN 118 (5.4%) 41 (5.3%) 19 (3.0%) 42 (7.1%) 16 (9.6%) HBV RNA Ever measured during 2001-2010a 1510 (68.6%) 449 (57.9%) 464 (71.1%) 505 (83.3%) 92 (55.1%) Most recent viral load during2001–2010a,b, IU/mL Undetectable 706 (46.8%) 148 (33.0%) 224 (48.3%) 283 (56.0%) 51 (55.4%) >300 to ≤2000 (detectable) 274 (18.1%) 100 (22.3%) 101 (21.8%) 62 (12.3%) 11 (12.0%) >2000 to ≤20 000 209 (13.8%) 76 (16.9%) 66 (14.2%) 66 (13.1%) 1 (1.1%) >20 000 301 (19.9%) 125 (27.8%) 73 (15.7%) 94 (18.6%) 9 (9.8%) E-antigen Ever tested during 2001–2010a,c 1471 (66.8%) 492 (63.4%) 448 (68.6%) 422 (69.6%) 109 (65.3%) Ever e-antigen positive 431 (29.3%) 145 (29.5%) 101 (22.5%) 143 (33.9%) 42 (38.5%) Always e-antigen negative 1035 (70.4%) 345 (70.1%) 346 (77.2%) 277 (65.6%) 67 (61.5%) Most recent e-antigen positive 338 (23.0%) 118 (24.0%) 82 (18.3%) 108 (25.6%) 30 (27.5%) Most recent e-antigen negative 1128 (76.7%) 371 (75.4%) 364 (81.3%) 314 (74.4%) 79 (72.5%) Liver biopsy at least once during 2001–2010a 490 (22.3%) 151 (19.5%) 130 (19.9%) 169 (27.9%) 40 (24.0%) HIV coinfected 142 (6.4%) 54 (7.0%) 17 (2.6%) 60 (9.9%) 11 (6.6%) Ever tested for HIVa 697 (31.7%) 229 (29.5%) 221 (33.8%) 234 (38.6%) 13 (7.8%) Hospitalized Ever hospitalized during 2001–2010 835 (37.9%) 299 (38.5%) 266 (40.7%) 210 (34.7%) 60 (35.9%) Ever had liver transplant, through 2010a 46 (2.1%) 6 (0.8%) 5 (0.8%) 32 (5.3%) 3 (1.8%) Deaths, 2006–2010 Died 2006–2010 199 (9.0%) 38 (4.9%) 54 (8.3%) 83 (13.7%) 24 (14.4%) Death rate per 1000 py (95% CI) 21.6 (15.5 – 27.7) 11.2 (3.8 – 18.7) 19.9 (9.2 – 30.7) 32.6 (18.5 – 46.7) 41.5 (11.3 – 71.8) No. of deaths by birth year cohort (years of age in 2008) and death rate per 1000 py (95% CI) 1985–1990 (18–23 y) 1975–1984 (24–33 y) 3 (0.9%) Death rate 2.2 (0–7.1) 1965–1974 (34–43 y) 22 (4.4%) Death rate 10.2 (1.4–19.1) 1955–1964 (44–53 y) 44 (7.3%) Death rate 17.0 (6.7–27.3) 1945–1954 (54–63 y) 69 (15.3%) Death rate 37.1 (19.7–54.5) 1935–1944 (64–73 y) 33 (18.6%) Death rate 46.5 (15.4–77.5) ≤1934 (≥74 y) 28 (35.9%) Death rate 99.7 (33.2–166.1) Characteristic Total (N = 2202) Portland, Oregon (n = 776; 35.2%) Hawaii (n = 653; 29.7%) Detroit, Michigan (n = 606; 27.5%) Danville, Pennsylvania (n = 167; 7.6%) Demographic Characteristic Birth year cohort (age in 2008) 1985–1990 (18–23 y) 54 (2.5%) 19 (2.4%) 20 (3.1%) 10 (1.7%) 5 (3.0%) 1975–1984 (24–33 y) 338 (15.3%) 128 (16.5%) 114 (17.5%) 62 (10.2%) 34 (20.4%) 1965–1974 (34–43 y) 501 (22.8%) 192 (24.7%) 130 (19.9%) 147 (24.3%) 32 (19.2%) 1955–1964 (44–53 y) 602 (27.3%) 215 (27.7%) 159 (24.3%) 183 (30.2%) 45 (26.9%) 1945–1954 (54–63 y) 452 (20.5%) 148 (19.1%) 132 (20.2%) 139 (22.9%) 33 (19.8%) 1935–1944 (64–73 y) 177 (8.0%) 55 (7.1%) 68 (10.4%) 47 (7.8%) 7 (4.2%) ≤1934 (≥74 y) 78 (3.5%) 19 (2.4%) 30 (4.6%) 18 (3.0%) 11 (6.6%) Sex Female 954 (43.3%) 338 (43.6%) 331 (50.7%) 223 (36.8%) 62 (37.1%) Male 1248 (56.7%) 438 (56.4%) 322 (49.3%) 383 (63.2%) 105 (62.9%) Race/ethnicity Race status known 1803 (81.9%) 584 (75.3%) 518 (79.3%) 546 (90.1%) 155 (92.8%) White 510 (28.3%) 198 (33.9%) 18 (3.5%) 187 (34.2%) 107 (69.0%) Black 239 (13.3%) 23 (3.9%) 5 (1.0%) 197 (36.1%) 14 (9.0%) Asian 843 (46.8%) 345 (59.1%) 309 (59.7%) 155 (28.4%) 34 (21.9%) Pacific Islander 198 (11.0%) 15 (2.6%) 183 (35.3%) American Indian/Alaska Native 13 (0.7%) 3 (0.5%) 3 (0.6%) 7 (1.3%) Ethnicity status known 1706 (77.5%) 493 (63.5%) 480 (73.5%) 566 (93.4%) 167 (100.0%) Hispanic ethnicity 39 (2.3%) 10 (2.0%) 11 (2.3%) 6 (1.1%) 12 (7.2%) Estimated household income (census tract geocode) Status known 2132 (96.8%) 767 (98.8%) 633 (96.9%) 575 (94.9%) 157 (94.0%) <$15 000 33 (1.5%) 4 (0.5%) 4 (0.6%) 17 (3.0%) 8 (5.1%)
$15 000–$29 000 258 (12.1%) 55 (7.2%) 63 (10.0%) 99 (17.2%) 41 (26.1%)
$30 000–$49 000 904 (42.4%) 370 (48.2%) 263 (41.5%) 180 (31.3%) 91 (58.0%)
$50 000–$75 000 674 (31.6%) 260 (33.9%) 248 (39.2%) 150 (26.1%) 16 (10.2%)
>$75 000 263 (12.3%) 78 (10.2%) 55 (8.7%) 129 (22.4%) 1 (0.6%) Insurance Status known 2096 (95.2%) 698 (89.9%) 625 (95.7%) 606 (100.0%) 167 (100.0%) Medicaid 107 (5.1%) 12 (1.7%) 36 (5.8%) 36 (5.9%) 23 (13.8%) Medicare only 23 (1.1%) 23 (13.8%) Medicare Plus 323 (15.4%) 70 (10.0%) 90 (14.4%) 163 (26.9%) Private 1600 (76.3%) 616 (88.3%) 499 (79.8%) 383 (63.2%) 102 (61.1%) None 43 (2.1%) 24 (4.0%) 19 (11.4%) Clinical or Laboratory Characteristic ALT Ever measured during 2001-2010a 2173 (98.7%) 768 (99.0%) 644 (98.6%) 595 (98.2%) 166 (99.4%) Highest ever ALT in 2001–2010a Less than ULN 1103 (50.8%) 401 (52.2%) 397 (61.6%) 237 (39.8%) 68 (41.0%) >ULN to ≤2 × ULN 457 (21.0%) 171 (22.3%) 123 (19.1%) 128 (21.5%) 35 (21.1%) >2 × ULN 613 (28.2%) 196 (25.5%) 124 (19.3%) 230 (38.7%) 63 (38.0%) Most recent ALT in 2001–2010a Less than ULN 1802 (82.9%) 624 (81.3%) 572 (88.8%) 485 (81.5%) 121 (72.9%) >ULN to ≤2 × ULN 253 (11.6%) 103 (13.4%) 53 (8.2%) 68 (11.4%) 29 (17.5%) >2 × ULN 118 (5.4%) 41 (5.3%) 19 (3.0%) 42 (7.1%) 16 (9.6%) HBV RNA Ever measured during 2001-2010a 1510 (68.6%) 449 (57.9%) 464 (71.1%) 505 (83.3%) 92 (55.1%) Most recent viral load during2001–2010a,b, IU/mL Undetectable 706 (46.8%) 148 (33.0%) 224 (48.3%) 283 (56.0%) 51 (55.4%) >300 to ≤2000 (detectable) 274 (18.1%) 100 (22.3%) 101 (21.8%) 62 (12.3%) 11 (12.0%) >2000 to ≤20 000 209 (13.8%) 76 (16.9%) 66 (14.2%) 66 (13.1%) 1 (1.1%) >20 000 301 (19.9%) 125 (27.8%) 73 (15.7%) 94 (18.6%) 9 (9.8%) E-antigen Ever tested during 2001–2010a,c 1471 (66.8%) 492 (63.4%) 448 (68.6%) 422 (69.6%) 109 (65.3%) Ever e-antigen positive 431 (29.3%) 145 (29.5%) 101 (22.5%) 143 (33.9%) 42 (38.5%) Always e-antigen negative 1035 (70.4%) 345 (70.1%) 346 (77.2%) 277 (65.6%) 67 (61.5%) Most recent e-antigen positive 338 (23.0%) 118 (24.0%) 82 (18.3%) 108 (25.6%) 30 (27.5%) Most recent e-antigen negative 1128 (76.7%) 371 (75.4%) 364 (81.3%) 314 (74.4%) 79 (72.5%) Liver biopsy at least once during 2001–2010a 490 (22.3%) 151 (19.5%) 130 (19.9%) 169 (27.9%) 40 (24.0%) HIV coinfected 142 (6.4%) 54 (7.0%) 17 (2.6%) 60 (9.9%) 11 (6.6%) Ever tested for HIVa 697 (31.7%) 229 (29.5%) 221 (33.8%) 234 (38.6%) 13 (7.8%) Hospitalized Ever hospitalized during 2001–2010 835 (37.9%) 299 (38.5%) 266 (40.7%) 210 (34.7%) 60 (35.9%) Ever had liver transplant, through 2010a 46 (2.1%) 6 (0.8%) 5 (0.8%) 32 (5.3%) 3 (1.8%) Deaths, 2006–2010 Died 2006–2010 199 (9.0%) 38 (4.9%) 54 (8.3%) 83 (13.7%) 24 (14.4%) Death rate per 1000 py (95% CI) 21.6 (15.5 – 27.7) 11.2 (3.8 – 18.7) 19.9 (9.2 – 30.7) 32.6 (18.5 – 46.7) 41.5 (11.3 – 71.8) No. of deaths by birth year cohort (years of age in 2008) and death rate per 1000 py (95% CI) 1985–1990 (18–23 y) 1975–1984 (24–33 y) 3 (0.9%) Death rate 2.2 (0–7.1) 1965–1974 (34–43 y) 22 (4.4%) Death rate 10.2 (1.4–19.1) 1955–1964 (44–53 y) 44 (7.3%) Death rate 17.0 (6.7–27.3) 1945–1954 (54–63 y) 69 (15.3%) Death rate 37.1 (19.7–54.5) 1935–1944 (64–73 y) 33 (18.6%) Death rate 46.5 (15.4–77.5) ≤1934 (≥74 y) 28 (35.9%) Death rate 99.7 (33.2–166.1) Data are presented as No. (%) unless otherwise specified. Abbreviations: ALT, alanine aminotransferase; CI, confidence interval; HBV, hepatitis B virus; HIV, human immunodeficiency virus; py, person-years; ULN, upper limit of normal. a Data on these specific variables available through 2008 in Portland. b In Danville, 20 (21.7%) additional patients had a detectable viral load in units other than IU/mL. c Indeterminate e-antigen results for 5 patients not shown in table. Among the 706 (46.8%) cohort patients whose most recent quantitative HBV DNA was “undetectable” (Table 1), 409 (57.9%) had documentation of antiviral treatment. During 2001–2010, 22.3% of the HBV patients had a liver biopsy and 37.9% of cohort patients were hospitalized once or more (Table 1); on average since 2006 >9% were hospitalized each year. A number of patients (2.1% overall, range, 0.4% [Hawaii and Portland] to 5.3% [Detroit]) had undergone liver transplant for end-stage liver disease by the end of 2010 (Table 1). Data from the Kaiser sites on transplants performed at referral facilities are captured via claims data that were included in the analysis. Nine percent of these HBV cohort patients, who all had at least 1 healthcare encounter during 2006–2008, had died (any cause) by the end of 2010. Of these 199 deaths, more than half (57%) occurred among persons born between 1945 and 1964 (Table 1). ### Chronic Hepatitis C Cohort We examined the records of 1 652 055 patients aged ≥18 years in the 4 participating health systems with encounters during 2006–2008. Of note, we found that of 9086 patients with a positive HCV antibody test during 2006–2008, 3428 (37.7%) had no documented follow-up HCV RNA testing in the electronic database. A total of 9933 patients met the electronic hepatitis C cohort inclusion criteria, and after exclusion of patients for whom chronic infection was not confirmed by abstractor records review, 8810 (88.7%) remained in the cohort (Figure 2). Median observation time was 5.3 years (range, <1–17 years), with a total of 58 147 person-years of observation. Demographic and clinical features of the 8810 confirmed chronic hepatitis C cohort patients are shown in Table 2. Across all sites, almost three-quarters of the patients were born between 1945 and 1964 (Table 2). Racial and ethnic mix varied greatly by study site, ranging from 3.2% black in Hawaii to 51.6% in Detroit, and from <1% Asian/Pacific Islander in Danville to 37.2% in Hawaii. Household income was less than$30 000/year for 21.5% overall, ranging from 10.5% in Portland to 33% in Danville. Payer also varied by site; the percentage of patients using only public insurance (Medicaid or Medicare only) ranged from 2.3% in Portland to 51.0% in Danville; also, at the Danville and Detroit sites, 7.6% and 4.3% of patients respectively, were uninsured (Table 2).

Table 2.

Demographic, Clinical, and Laboratory Characteristics of Patients With Chronic Hepatitis C

Total (N = 8810) Portland, Oregon (n = 2955; 33.5%) Hawaii (n = 1040; 11.8%) Detroit, Michigan (n = 3067; 34.8%) Danville, Pennsylvania (n = 1748; 19.8%)
Demographic Characteristic
Birth year cohort (age in 2008)
1985–1990 (18–23 y) 121 (1.4%) 14 (0.5%) 17 (0.6%) 90 (5.1%)
1975–1984 (24–33 y) 407 (4.6%) 71 (2.4%) 7 (0.7%) 48 (1.6%) 281 (16.1%)
1965–1974 (34–43 y) 724 (8.2%) 264 (8.9%) 52 (5.0%) 162 (5.3%) 246 (14.1%)
1955–1964 (44–53 y) 2936 (33.3%) 1108 (37.5%) 364 (35.0%) 816 (26.6%) 648 (37.1%)
1945–1954 (54–63 y) 3704 (42.0%) 1260 (42.6%) 490 (47.1%) 1554 (50.7%) 400 (22.9%)
1935–1944 (64–73 y) 633 (7.2%) 172 (5.8%) 91 (8.8%) 322 (10.5%) 48 (2.7%)
≤1934 (≥74 y) 285 (3.2%) 66 (2.2%) 36 (3.5%) 148 (4.8%) 35 (2.0%)
Sex
Female 3554 (40.3%) 1216 (41.2%) 360 (34.6%) 1180 (38.5%) 798 (45.7%)
Male 5256 (59.7%) 1739 (58.8%) 680 (65.4%) 1887 (61.5%) 950 (54.3%)
Race/ethnicity
Race status known 7710 (87.5%) 2330 (78.8%) 784 (75.4%) 2893 (94.3%) 1703 (97.4%)
White 5361 (69.5%) 2021 (86.7%) 438 (55.9%) 1316 (45.5%) 1586 (93.1%)
Black 1751 (22.7%) 141 (6.1%) 25 (3.2%) 1493 (51.6%) 92 (5.4%)
Asian 267 (3.5%) 77 (3.3%) 122 (15.6%) 62 (2.1%) 6 (0.4%)
Pacific Islander 185 (2.4%) 14 (0.6%) 170 (21.7%) 1 (0.1%)
American Indian/Alaska Native 146 (1.9%) 77 (3.3%) 29 (3.7%) 22 (0.8%) 18 (1.1%)
Ethnicity status known 7481 (84.9%) 2022 (68.4%) 764 (73.5%) 2951 (96.2%) 1744 (99.8%)
Hispanic ethnicity 304 (4.1%) 121 (6.0%) 75 (9.8%) 51 (1.7%) 57 (3.3%)
Estimated household income (census tract geocode)
Status known 8576 (97.3%) 2940 (99.5%) 998 (96.0%) 2949 (96.2%) 1689 (96.6%)
<$15 000 198 (2.3%) 16 (0.5%) 8 (0.8%) 118 (4.0%) 56 (3.3%)$15 000–$29 000 1649 (19.2%) 292 (9.9%) 99 (9.9%) 761 (25.8%) 497 (29.4%)$30 000–$49 000 4150 (48.4%) 1575 (53.6%) 456 (45.7%) 1076 (36.5%) 1043 (61.8%)$50 000–$75 000 2064 (24.1%) 938 (31.9%) 349 (35.0%) 686 (23.3%) 91 (5.4%) >$75 000 515 (6.0%) 119 (4.0%) 86 (8.6%) 308 (10.4%) 2 (0.1%)
Insurance
Status known 8424 (95.6%) 2657 (89.9%) 952 (91.5%) 3067 (100.0%) 1748 (100.0%)
Medicaid 1021 (12.1%) 61 (2.3%) 112 (11.8%) 278 (9.1%) 570 (32.6%)
Medicare only 322 (3.8%) 322 (18.4%)
Medicare Plus 1611 (19.1%) 292 (11.0%) 152 (16.0%) 1167 (38.1%)
Private 5204 (61.8%) 2304 (86.7%) 688 (72.3%) 1489 (48.5%) 723 (41.4%)
None 266 (3.2%) 133 (4.3%) 133 (7.6%)
Laboratory or Clinical Characteristic
ALT
Ever measured during  2001–2010a 8674 (98.5%) 2924 (99.0%) 1030 (99.0%) 2997 (97.7%) 1723 (98.6%)
Highest ever ALT in 2001–2010a
Less than ULN 1787 (20.6%) 685 (23.4%) 296 (28.7%) 467 (15.6%) 339 (19.7%)
>ULN to ≤2 × ULN 2687 (31.0%) 890 (30.4%) 312 (30.3%) 980 (32.7%) 505 (29.3%)
>2 × ULN 4200 (48.4%) 1349 (46.1%) 422 (41.0%) 1550 (51.7%) 879 (51.0%)
Most recent ALT in 2001–2010a
Less than ULN 5056 (58.3%) 1622 (55.5%) 633 (61.5%) 1828 (61.0%) 973 (56.5%)
>ULN to ≤2 × ULN 2282 (26.3%) 806 (27.6%) 255 (24.8%) 773 (25.8%) 448 (26.0%)
>2 × ULN 1336 (15.4%) 496 (17.0%) 142 (13.8%) 396 (13.2%) 302 (17.5%)
HCV RNA
Ever measured during  2001–2010a 5540 (62.9%) 714 (24.2%) 898 (86.3%) 2414 (78.7%) 1514 (86.6%)
Most recent viral load during 2001–2010a, IU/mL
Undetectable 1175 (21.2%) 269 (37.7%) 179 (19.9%) 385 (15.9%) 342 (22.6%)
Detectable but ≤100 000 644 (11.6%) 70 (9.8%) 100 (11.1%) 244 (10.1%) 230 (15.2%)
>100 000 3721 (67.2%) 375 (52.5%) 619 (68.9%) 1785 (73.9%) 942 (62.2%)
Liver biopsy at least once during 2001–2010a 3380 (38.4%) 1159 (39.2%) 436 (41.9%) 1089 (35.5%) 696 (39.8%)
HIV coinfected 258 (2.9%) 69 (2.3%) 15 (1.4%) 143 (4.7%) 31 (1.8%)
Ever tested for HIVa 2436 (27.7%) 665 (22.5%) 481 (46.3%) 1245 (40.6%) 45 (2.6%)
Hospitalized
Ever hospitalized during  2001–2010 3899 (44.3%) 1281 (43.4%) 422 (40.6%) 1561 (50.9%) 635 (36.3%)
Ever had liver transplant, through 2010a 415 (4.7%) 35 (1.2%) 18 (1.7%) 319 (10.4%) 43 (2.5%)
Deaths, 2006–2010
Died 2006–2010 1229 (14.0%) 191 (6.5%) 113 (10.9%) 710 (23.1%) 215 (12.3%)
Death rate per 1000 py (95% CI) 33.0 (29.2–36.7) 14.1 (9.8–18.3) 24.9 (15.4–34.3) 57.2 (49.0–65.4) 31.9 (23.6–40.1)
No. of deaths by birth year cohort (years of age in 2008) and death rate per 1000 py (95% CI)
1985–1990 (18–23 y) 4 (3.3%)
Death rate 8.3 (0–24.5)
1975–1984 (24–33 y) 11 (2.7%)
Death rate 6.6 (0.0–14.5)
1965–1974 (34–43 y) 38 (5.2%)
Death rate 12.4 (4.3–20.4)
1955–1964 (44–53 y) 287 (9.8%)
Death rate 22.8 (17.4–28.2)
1945–1954 (54-63 y) 621 (16.8%)
Death rate 39.3 (33.1–45.6)
1935–1944 (64–73 y) 150 (23.7%)
Death rate 57.0 (39.0–75.1)
≤1934 (≥74 y) 118 (41.4%)
Death rate 110.8 (74.4–147.3)
Total (N = 8810) Portland, Oregon (n = 2955; 33.5%) Hawaii (n = 1040; 11.8%) Detroit, Michigan (n = 3067; 34.8%) Danville, Pennsylvania (n = 1748; 19.8%)
Demographic Characteristic
Birth year cohort (age in 2008)
1985–1990 (18–23 y) 121 (1.4%) 14 (0.5%) 17 (0.6%) 90 (5.1%)
1975–1984 (24–33 y) 407 (4.6%) 71 (2.4%) 7 (0.7%) 48 (1.6%) 281 (16.1%)
1965–1974 (34–43 y) 724 (8.2%) 264 (8.9%) 52 (5.0%) 162 (5.3%) 246 (14.1%)
1955–1964 (44–53 y) 2936 (33.3%) 1108 (37.5%) 364 (35.0%) 816 (26.6%) 648 (37.1%)
1945–1954 (54–63 y) 3704 (42.0%) 1260 (42.6%) 490 (47.1%) 1554 (50.7%) 400 (22.9%)
1935–1944 (64–73 y) 633 (7.2%) 172 (5.8%) 91 (8.8%) 322 (10.5%) 48 (2.7%)
≤1934 (≥74 y) 285 (3.2%) 66 (2.2%) 36 (3.5%) 148 (4.8%) 35 (2.0%)
Sex
Female 3554 (40.3%) 1216 (41.2%) 360 (34.6%) 1180 (38.5%) 798 (45.7%)
Male 5256 (59.7%) 1739 (58.8%) 680 (65.4%) 1887 (61.5%) 950 (54.3%)
Race/ethnicity
Race status known 7710 (87.5%) 2330 (78.8%) 784 (75.4%) 2893 (94.3%) 1703 (97.4%)
White 5361 (69.5%) 2021 (86.7%) 438 (55.9%) 1316 (45.5%) 1586 (93.1%)
Black 1751 (22.7%) 141 (6.1%) 25 (3.2%) 1493 (51.6%) 92 (5.4%)
Asian 267 (3.5%) 77 (3.3%) 122 (15.6%) 62 (2.1%) 6 (0.4%)
Pacific Islander 185 (2.4%) 14 (0.6%) 170 (21.7%) 1 (0.1%)
American Indian/Alaska Native 146 (1.9%) 77 (3.3%) 29 (3.7%) 22 (0.8%) 18 (1.1%)
Ethnicity status known 7481 (84.9%) 2022 (68.4%) 764 (73.5%) 2951 (96.2%) 1744 (99.8%)
Hispanic ethnicity 304 (4.1%) 121 (6.0%) 75 (9.8%) 51 (1.7%) 57 (3.3%)
Estimated household income (census tract geocode)
Status known 8576 (97.3%) 2940 (99.5%) 998 (96.0%) 2949 (96.2%) 1689 (96.6%)
<$15 000 198 (2.3%) 16 (0.5%) 8 (0.8%) 118 (4.0%) 56 (3.3%)$15 000–$29 000 1649 (19.2%) 292 (9.9%) 99 (9.9%) 761 (25.8%) 497 (29.4%)$30 000–$49 000 4150 (48.4%) 1575 (53.6%) 456 (45.7%) 1076 (36.5%) 1043 (61.8%)$50 000–$75 000 2064 (24.1%) 938 (31.9%) 349 (35.0%) 686 (23.3%) 91 (5.4%) >$75 000 515 (6.0%) 119 (4.0%) 86 (8.6%) 308 (10.4%) 2 (0.1%)
Insurance
Status known 8424 (95.6%) 2657 (89.9%) 952 (91.5%) 3067 (100.0%) 1748 (100.0%)
Medicaid 1021 (12.1%) 61 (2.3%) 112 (11.8%) 278 (9.1%) 570 (32.6%)
Medicare only 322 (3.8%) 322 (18.4%)
Medicare Plus 1611 (19.1%) 292 (11.0%) 152 (16.0%) 1167 (38.1%)
Private 5204 (61.8%) 2304 (86.7%) 688 (72.3%) 1489 (48.5%) 723 (41.4%)
None 266 (3.2%) 133 (4.3%) 133 (7.6%)
Laboratory or Clinical Characteristic
ALT
Ever measured during  2001–2010a 8674 (98.5%) 2924 (99.0%) 1030 (99.0%) 2997 (97.7%) 1723 (98.6%)
Highest ever ALT in 2001–2010a
Less than ULN 1787 (20.6%) 685 (23.4%) 296 (28.7%) 467 (15.6%) 339 (19.7%)
>ULN to ≤2 × ULN 2687 (31.0%) 890 (30.4%) 312 (30.3%) 980 (32.7%) 505 (29.3%)
>2 × ULN 4200 (48.4%) 1349 (46.1%) 422 (41.0%) 1550 (51.7%) 879 (51.0%)
Most recent ALT in 2001–2010a
Less than ULN 5056 (58.3%) 1622 (55.5%) 633 (61.5%) 1828 (61.0%) 973 (56.5%)
>ULN to ≤2 × ULN 2282 (26.3%) 806 (27.6%) 255 (24.8%) 773 (25.8%) 448 (26.0%)
>2 × ULN 1336 (15.4%) 496 (17.0%) 142 (13.8%) 396 (13.2%) 302 (17.5%)
HCV RNA
Ever measured during  2001–2010a 5540 (62.9%) 714 (24.2%) 898 (86.3%) 2414 (78.7%) 1514 (86.6%)
Most recent viral load during 2001–2010a, IU/mL
Undetectable 1175 (21.2%) 269 (37.7%) 179 (19.9%) 385 (15.9%) 342 (22.6%)
Detectable but ≤100 000 644 (11.6%) 70 (9.8%) 100 (11.1%) 244 (10.1%) 230 (15.2%)
>100 000 3721 (67.2%) 375 (52.5%) 619 (68.9%) 1785 (73.9%) 942 (62.2%)
Liver biopsy at least once during 2001–2010a 3380 (38.4%) 1159 (39.2%) 436 (41.9%) 1089 (35.5%) 696 (39.8%)
HIV coinfected 258 (2.9%) 69 (2.3%) 15 (1.4%) 143 (4.7%) 31 (1.8%)
Ever tested for HIVa 2436 (27.7%) 665 (22.5%) 481 (46.3%) 1245 (40.6%) 45 (2.6%)
Hospitalized
Ever hospitalized during  2001–2010 3899 (44.3%) 1281 (43.4%) 422 (40.6%) 1561 (50.9%) 635 (36.3%)
Ever had liver transplant, through 2010a 415 (4.7%) 35 (1.2%) 18 (1.7%) 319 (10.4%) 43 (2.5%)
Deaths, 2006–2010
Died 2006–2010 1229 (14.0%) 191 (6.5%) 113 (10.9%) 710 (23.1%) 215 (12.3%)
Death rate per 1000 py (95% CI) 33.0 (29.2–36.7) 14.1 (9.8–18.3) 24.9 (15.4–34.3) 57.2 (49.0–65.4) 31.9 (23.6–40.1)
No. of deaths by birth year cohort (years of age in 2008) and death rate per 1000 py (95% CI)
1985–1990 (18–23 y) 4 (3.3%)
Death rate 8.3 (0–24.5)
1975–1984 (24–33 y) 11 (2.7%)
Death rate 6.6 (0.0–14.5)
1965–1974 (34–43 y) 38 (5.2%)
Death rate 12.4 (4.3–20.4)
1955–1964 (44–53 y) 287 (9.8%)
Death rate 22.8 (17.4–28.2)
1945–1954 (54-63 y) 621 (16.8%)
Death rate 39.3 (33.1–45.6)
1935–1944 (64–73 y) 150 (23.7%)
Death rate 57.0 (39.0–75.1)
≤1934 (≥74 y) 118 (41.4%)
Death rate 110.8 (74.4–147.3)

Data are presented as No. (%) unless otherwise specified.

Abbreviations: ALT, alanine aminotransferase; CI, confidence interval; HCV, hepatitis C virus; HIV, human immunodeficiency virus; py, person-years; ULN, upper limit of normal.

a Data on these specific variables available through 2008 in Portland.

Among the 5540 patients (62.9% overall, range by site, 24.2%–86.6%) with quantitative HCV RNA tests during 2001–2010, recent values were undetectable for 1175 (21.2%; Table 1), of whom 937 (79.7%) patients had documentation of having received antiviral treatment. Overall in the cohort, 3211 (36.5%) had documentation of attempted antiviral treatment. During 2001–2010, 38.4% of the chronic hepatitis C patients had a liver biopsy, and 28% had ever been tested for HIV, with 2.9% testing positive. During 2001–2010, 44.3% experienced at least 1 hospitalization, and on average since 2006 >13% were hospitalized each year. Many patients (4.7%, ranging from 1.2% in Portland to 10.4% in Detroit) had undergone liver transplant for end-stage liver disease by 31 December 2010.

Fourteen percent of these HCV cohort patients, who all had at least 1 healthcare encounter during 2006–2008, had died (any cause) by the end of 2010 (Table 2). Most deaths (n = 908, 73.9%) occurred among persons in the 1945–1964 birth cohort, with an overall death rate of 33.0 per 1000 person-years (Table 2). Table 3 shows the elevated death rates per 1000 person-years in the CHeCS HCV cohort as compared with HCV antibody–negative persons in NHANES. These mortality rates were significantly elevated overall, when stratified by sex, and when stratified by age among persons aged 30–59 years.

Table 3.

All-Cause Death Rates in the CHeCS Chronic Hepatitis C Virus (HCV) Patients, 2006–2010, Compared With HCV Antibody–Negative Adults in NHANES III (1988–1994)

Deaths in CHeCSHCV Cohort, No. (%) Death Rate per 1000 pyin CHeCS Death Rate per 1000 py for HCV Antibody–Negative Adults in NHANES IIIa
All participants 1229 (100) 33.0 (29.2–36.7)c 11.1 (10.1–12.2)c
Men 836 (68.0) 37.9 (32.8–43.1)c 11.6 (10.5–12.7)c
17–29 yb 5 (0.4) 10.4 (0–28.7) 1.3 (1.5–3.4)
30–44 y 27 (2.2) 15.3 (3.7–27.0)c 2.3 (1.6–3.1)c
45–59 y 525 (42.7) 39.1 (32.4–45.8)c 11.2 (9.1–13.7)c
≥60 y 279 (22.7) 36.9 (24.9–48.9) 54.2 (49.3–59.6)
Women 393 (32.0) 25.8 (20.6–31.0)c 10.6 (9.6–11.8)c
17–29 ya 8 (0.7) 8.0 (0–8.8) 1.2 (0.8–2.0)
30–44 y 25 (2.0) 13.4 (2.7–24.0)c 1.8 (1.2–2.6)c
45–59 y 209 (17.0) 25.0 (18.1–32.0)c 8.5 (6.8–10.6)c
≥60 y 151 (12.3) 47.7 (32.7–62.7) 43.4 (39.2–47.9)
Deaths in CHeCSHCV Cohort, No. (%) Death Rate per 1000 pyin CHeCS Death Rate per 1000 py for HCV Antibody–Negative Adults in NHANES IIIa
All participants 1229 (100) 33.0 (29.2–36.7)c 11.1 (10.1–12.2)c
Men 836 (68.0) 37.9 (32.8–43.1)c 11.6 (10.5–12.7)c
17–29 yb 5 (0.4) 10.4 (0–28.7) 1.3 (1.5–3.4)
30–44 y 27 (2.2) 15.3 (3.7–27.0)c 2.3 (1.6–3.1)c
45–59 y 525 (42.7) 39.1 (32.4–45.8)c 11.2 (9.1–13.7)c
≥60 y 279 (22.7) 36.9 (24.9–48.9) 54.2 (49.3–59.6)
Women 393 (32.0) 25.8 (20.6–31.0)c 10.6 (9.6–11.8)c
17–29 ya 8 (0.7) 8.0 (0–8.8) 1.2 (0.8–2.0)
30–44 y 25 (2.0) 13.4 (2.7–24.0)c 1.8 (1.2–2.6)c
45–59 y 209 (17.0) 25.0 (18.1–32.0)c 8.5 (6.8–10.6)c
≥60 y 151 (12.3) 47.7 (32.7–62.7) 43.4 (39.2–47.9)

Abbreviations: CHeCS, Chronic Hepatitis Cohort Study; HCV, hepatitis C virus; NHANES, National Health and Nutrition Examination Survey; py, person-years.

a Among NHANES III participants 1988–1994 with mortality follow-up through 2006 [7].

b Years of age at time of death or last observation.

c Statistically significant difference (P < .05) between CHeCS and NHANES populations.

## DISCUSSION

This is the first US study to characterize a general population of >11 000 patients with chronic HBV and HCV infections and describe their substantial rates of morbidity and mortality. Rates of hospitalization were considerable; approximately 40% of both cohorts had experienced 1 or more hospitalizations during 2001–2010, with 9% of the HBV and 13% of the HCV cohorts hospitalized in an average year. Many patients (4.7% of HCV and 2.1% of HBV cohorts) had undergone liver transplant for end-stage liver disease; health systems included liver transplant centers in Detroit (since 1989) and Danville (since 2006).

The mortality rates and the relative youth of CHeCS patients who were dying were also notable. As in a recent study of 22 million death certificates from the period 1999–2007 [8], most deaths occurred among HBV- and HCV-infected persons aged 45–64 years. In particular, we found all-cause death rates among the HCV-infected CHeCS patients to be 3 times higher than those recorded among HCV-uninfected NHANES patients [7] (33.0 vs 11.1 deaths per 1000 person-years). This may partly result from the fact that the transplant centers in Detroit and Danville may augment the inclusion of patients with advanced liver disease in the CHeCS. However, the rates and percentages of deaths at the 2 nontransplant sites during the 5 years of study (11.2–19.9 deaths per 1000 person-years [4.9%–8.3%] for HBV and 14.1–24.9 deaths per 1000 person-years [6.5%–10.9%] for HCV patients) were also substantial. In summary, these high hospitalization and mortality rates, especially among those aged 30–60 years, are remarkable and worrisome.

Data from CHeCS are already yielding unique and useful information that may be used for public health action and policy development. For example, given the prevalence of infection and deaths in HCV patients in the CHeCS, our results strongly support the new policy of “birth cohort” testing of those born in 1945 through 1964 [12]. This approach is likely to be more effective than the previous risk-based testing strategy in identifying persons infected with HCV in the distant past [13].

Therapies for chronic viral hepatitis are likely to improve substantially over the coming years. Treatment results in clinical practice do not always mirror those in the clinical trial setting, particularly among minority and low-income patients with many barriers to care [10]. New patients are being added to the CHeCS dynamic cohort and records of existing patients will be updated yearly: as part of ongoing data collection, we examined the records of >1 million health system patients with encounters during 2009–2010 and identified >1200 additional chronic HBV and 3000 additional chronic HCV patients for cohort inclusion. An ongoing survey of patients in the CHeCS—of which >4500 have been completed—will further allow analyses to be adjusted for various comorbidities and conditions, such as smoking, substance use, and psychosocial measures of well-being and function. The CHeCS will address, under nonclinical trial, “real-world” conditions, issues regarding the effectiveness of evolving therapies on mortality, disease progression, adverse events, and drug resistance in a managed care population. Previously, this picture had to be estimated and pieced together from disparate and sparse data sources [11].

The CHeCS case definition algorithms applied to electronic health plan data were able to detect a large number of confirmed chronic HBV and HCV cases. While inclusion criteria based on positive nucleic acid tests (and for HBV hepatitis B surface antigen) were highly specific (>90% for HBV and >96% for HCV), use of ICD-9–based inclusion categories among patients without positive laboratory findings was of limited reliability without supplemental chart review.

CHeCS patients cannot be fully representative of chronic hepatitis B and C patients in the United States. First, these patients necessarily represent only those known to the medical care systems. As another CHeCS analysis of >1 million adult patients served by the 4 study sites has found, it appears that >21% of HBV and >43% of HCV patients in the community remain undiagnosed [9]. Institutionalized populations and the homeless are also underrepresented, although this is a limitation of almost any large study. Further, patients at 2 of the 4 participating sites are members of a health maintenance organization (Kaiser), which increases the percentage of persons with health insurance in the CHeCS as a whole.

However, these limitations should be balanced against the purposeful inclusion of Kaiser Permanente sites to make sure there was a strong representation of persons of Asian and Pacific Islander ethnicity and high numbers of HBV-infected persons. Similarly, inclusion of the sites in Detroit and central Pennsylvania assured adequate representation of Africans and African Americans and persons in important risk groups, especially injection drug users. Thus, there is good representation of minorities, the uninsured and underinsured, and those of low household income such that specific analyses of them, or analyses that adjust for their representation in the general population, can be performed.

In summary, the CHeCS is a large, multicenter study of patients with chronic hepatitis B and C that has already generated unique information about the high morbidity and mortality in this population. It is expected to yield much data-driven information about the impact of therapies, comorbidities, and conditions on the general population with HBV and HCV in the future.

## Notes

Acknowledgments. The CHeCS Investigators include the following investigators and sites: Scott D. Holmberg, Eyasu H. Teshale, Philip R. Spradling, and Anne C. Moorman, Division of Viral Hepatitis, National Centers for HIV, Viral Hepatitis, STD, and TB Prevention, Centers for Disease Control and Prevention (CDC), Atlanta, Georgia; Stuart C. Gordon, David R. Nerenz, Mei Lu, Lois Lamerato, Loralee B. Rupp, Nonna Akkerman, Nancy J. Oja-Tebbe, Chad M. Cogan, and Dana Larkin, Henry Ford Health System, Detroit, Michigan; Joseph A. Boscarino, Zahra S. Daar, Robert E. Smith, Patrick J. Curry, Brandon D. Geise, and Joe B. Leader, Geisinger Health System, Danville, Pennsylvania; Cynthia C. Nakasato, Vinutha Vijayadeva, Kelly E. Sylva, John V. Parker, and Mark M. Schmidt, Kaiser Permanente, Honolulu, Hawaii; Emily M. Henkle, Tracy L. Dodge, Erin M. Keast, and Lois Drew, Kaiser Permanente Northwest, Portland, Oregon.

Disclaimer. Granting corporations did not have access to CHeCS data and did not contribute to data analysis or writing of the manuscript.

Financial support. CHeCS was funded by the CDC Foundation, which received grants from Abbott Laboratories; Genentech, a member of the Roche Group; Janssen Pharmaceutical Companies of Johnson & Johnson; and Vertex Pharmaceuticals.

Potential conflicts of interest. S. C. G. has received grant/research support from Abbott Pharmaceuticals, Anadys Pharmaceuticals, Bristol-Myers Squibb, Conatus, Eiger Biopharmaceuticals, Exalenz BioScience, Gilead Pharmaceuticals, GlaxoSmithKline, GlobeImmune, Intercept Pharmaceuticals, Merck, Roche Pharmaceuticals, Tibotec, Vertex Pharmaceuticals, and Zymogenetics; has served as a consultant for Achillion, Bristol-Myers Squibb, CVS Caremark, Gilead Pharmaceuticals, Merck, Salix Pharmaceuticals, Johnson & Johnson, and Vertex; has served on the data monitoring board for Tibotec; and has served as a speaker for Bayer, Gilead, Roche, Merck, and Vertex. J. A. B. has consulted for Pfizer and Janssen. All other authors report no potential conflicts.

All authors have submitted the ICMJE Form for Disclosure of Potential Conflicts of Interest. Conflicts that the editors consider relevant to the content of the manuscript have been disclosed.

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## Author notes

a
The CHeCS investigators are listed in the Acknowledgments.


text:
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Section 2-2 : The Limit

In the previous section we looked at a couple of problems and in both problems we had a function (slope in the tangent problem case and average rate of change in the rate of change problem) and we wanted to know how that function was behaving at some point $$x = a$$. At this stage of the game we no longer care where the functions came from and we no longer care if we’re going to see them down the road again or not. All that we need to know or worry about is that we’ve got these functions and we want to know something about them.

To answer the questions in the last section we choose values of $$x$$ that got closer and closer to $$x = a$$ and we plugged these into the function. We also made sure that we looked at values of $$x$$ that were on both the left and the right of $$x = a$$. Once we did this we looked at our table of function values and saw what the function values were approaching as $$x$$ got closer and closer to $$x = a$$ and used this to guess the value that we were after.

This process is called taking a limit and we have some notation for this. The limit notation for the two problems from the last section is,

$\mathop {\lim }\limits_{x \to 1} \frac{{2 - 2{x^2}}}{{x - 1}} = - 4\hspace{0.75in}\mathop {\lim }\limits_{t \to 5} \frac{{{t^3} - 6{t^2} + 25}}{{t - 5}} = 15$

In this notation we will note that we always give the function that we’re working with and we also give the value of $$x$$ (or $$t$$) that we are moving in towards.

In this section we are going to take an intuitive approach to limits and try to get a feel for what they are and what they can tell us about a function. With that goal in mind we are not going to get into how we actually compute limits yet. We will instead rely on what we did in the previous section as well as another approach to guess the value of the limits.

Both approaches that we are going to use in this section are designed to help us understand just what limits are. In general, we don’t typically use the methods in this section to compute limits and in many cases can be very difficult to use to even estimate the value of a limit and/or will give the wrong value on occasion. We will look at actually computing limits in a couple of sections.

Let’s first start off with the following “definition” of a limit.

Definition

We say that the limit of $$f(x)$$ is $$L$$ as $$x$$ approaches $$a$$ and write this as

$\mathop {\lim }\limits_{x \to a} f\left( x \right) = L$

provided we can make $$f(x)$$ as close to $$L$$ as we want for all $$x$$ sufficiently close to $$a$$, from both sides, without actually letting $$x$$ be $$a$$.

This is not the exact, precise definition of a limit. If you would like to see the more precise and mathematical definition of a limit you should check out the The Definition of a Limit section at the end of this chapter. The definition given above is more of a “working” definition. This definition helps us to get an idea of just what limits are and what they can tell us about functions.

So just what does this definition mean? Well let’s suppose that we know that the limit does in fact exist. According to our “working” definition we can then decide how close to $$L$$ that we’d like to make $$f(x)$$. For sake of argument let’s suppose that we want to make $$f(x)$$ no more than 0.001 away from $$L$$. This means that we want one of the following

$\begin{array}{lcl}f\left( x \right) - L < 0.001 & \hspace{0.25in} & {\mbox{if }}f\left( x \right){\mbox{ is larger than L}}\\ L - f\left( x \right) < 0.001& \hspace{0.25in} & {\mbox{if }}f\left( x \right){\mbox{ is smaller than L}}\end{array}$

Now according to the “working” definition this means that if we get $$x$$ sufficiently close to $$a$$ we can make one of the above true. However, it actually says a little more. It says that somewhere out there in the world is a value of $$x$$, say $$X$$, so that for all $$x$$’s that are closer to $$a$$ than $$X$$ then one of the above statements will be true.

This is a fairly important idea. There are many functions out there in the world that we can make as close to $$L$$ for specific values of $$x$$ that are close to $$a$$, but there will be other values of $$x$$ closer to $$a$$ that give functions values that are nowhere near close to $$L$$. In order for a limit to exist once we get $$f(x)$$ as close to $$L$$ as we want for some $$x$$ then it will need to stay in that close to $$L$$ (or get closer) for all values of $$x$$ that are closer to $$a$$. We’ll see an example of this later in this section.

In somewhat simpler terms the definition says that as $$x$$ gets closer and closer to $$x=a$$ (from both sides of course…) then $$f(x)$$ must be getting closer and closer to $$L$$. Or, as we move in towards $$x=a$$ then $$f(x)$$ must be moving in towards $$L$$.

It is important to note once again that we must look at values of $$x$$ that are on both sides of $$x=a$$. We should also note that we are not allowed to use $$x=a$$ in the definition. We will often use the information that limits give us to get some information about what is going on right at $$x=a$$, but the limit itself is not concerned with what is actually going on at $$x=a$$. The limit is only concerned with what is going on around the point $$x=a$$. This is an important concept about limits that we need to keep in mind.

An alternative notation that we will occasionally use in denoting limits is

$f(x) \to L\hspace{0.25in}{\rm{as}}\hspace{0.25in}x \to a$

How do we use this definition to help us estimate limits? We do exactly what we did in the previous section. We take $$x$$’s on both sides of $$x=a$$ that move in closer and closer to $$a$$ and we plug these into our function. We then look to see if we can determine what number the function values are moving in towards and use this as our estimate.

Let’s work an example.

Example 1 Estimate the value of the following limit. $\mathop {\lim }\limits_{x \to 2} \frac{{{x^2} + 4x - 12}}{{{x^2} - 2x}}$
Show Solution

Notice that we did say estimate the value of the limit. Again, we are not going to directly compute limits in this section. The point of this section is to give us a better idea of how limits work and what they can tell us about the function.

So, with that in mind we are going to work this in pretty much the same way that we did in the last section. We will choose values of $$x$$ that get closer and closer to $$x=2$$ and plug these values into the function. Doing this gives the following table of values.

$$x$$ $$f(x)$$ $$x$$ $$f(x)$$
2.5 3.4 1.5 5.0
2.1 3.857142857 1.9 4.157894737
2.01 3.985074627 1.99 4.015075377
2.001 3.998500750 1.999 4.001500750
2.0001 3.999850007 1.9999 4.000150008
2.00001 3.999985000 1.99999 4.000015000

Note that we made sure and picked values of $$x$$ that were on both sides of $$x = 2$$ and that we moved in very close to $$x = 2$$ to make sure that any trends that we might be seeing are in fact correct.

Also notice that we can’t actually plug in $$x = 2$$ into the function as this would give us a division by zero error. This is not a problem since the limit doesn’t care what is happening at the point in question.

From this table it appears that the function is going to 4 as $$x$$ approaches 2, so

$\mathop {\lim }\limits_{x \to 2} \frac{{{x^2} + 4x - 12}}{{{x^2} - 2x}} = 4$

Let’s think a little bit more about what’s going on here. Let’s graph the function from the last example. The graph of the function in the range of $$x$$’s that were interested in is shown below.

First, notice that there is a rather large open dot at $$x = 2$$. This is there to remind us that the function (and hence the graph) doesn’t exist at $$x = 2$$.

As we were plugging in values of $$x$$ into the function we are in effect moving along the graph in towards the point as $$x = 2$$. This is shown in the graph by the two arrows on the graph that are moving in towards the point.

When we are computing limits the question that we are really asking is what $$y$$ value is our graph approaching as we move in towards $$x = a$$ on our graph. We are NOT asking what $$y$$ value the graph takes at the point in question. In other words, we are asking what the graph is doing around the point $$x = a$$. In our case we can see that as $$x$$ moves in towards 2 (from both sides) the function is approaching $$y = 4$$ even though the function itself doesn’t even exist at $$x = 2$$. Therefore, we can say that the limit is in fact 4.

So, what have we learned about limits? Limits are asking what the function is doing around $$x = a$$ and are not concerned with what the function is actually doing at $$x = a$$. This is a good thing as many of the functions that we’ll be looking at won’t even exist at $$x = a$$ as we saw in our last example.

Let’s work another example to drive this point home.

Example 2 Estimate the value of the following limit. $\mathop {\lim }\limits_{x \to 2} g\left( x \right)\hspace{0.25in}{\mbox{where,}}\hspace{0.25in}g\left( x \right) = \left\{ \begin{array}{ll}\displaystyle \frac{{{x^2} + 4x - 12}}{{{x^2} - 2x}} & {\mbox{if }}x \ne 2\\ 6 & {\mbox{if }}x = 2\end{array} \right.$
Show Solution

The first thing to note here is that this is exactly the same function as the first example with the exception that we’ve now given it a value for $$x = 2$$. So, let’s first note that

$g\left( 2 \right) = 6$

As far as estimating the value of this limit goes, nothing has changed in comparison to the first example. We could build up a table of values as we did in the first example or we could take a quick look at the graph of the function. Either method will give us the value of the limit.

Let’s first take a look at a table of values and see what that tells us. Notice that the presence of the value for the function at $$x = 2$$ will not change our choices for $$x$$. We only choose values of $$x$$ that are getting closer to $$x = 2$$ but we never take $$x = 2$$. In other words, the table of values that we used in the first example will be exactly the same table that we’ll use here. So, since we’ve already got it down once there is no reason to redo it here.

From this table it is again clear that the limit is,

$\mathop {\lim }\limits_{x \to 2} g\left( x \right) = 4$

The limit is NOT 6! Remember from the discussion after the first example that limits do not care what the function is actually doing at the point in question. Limits are only concerned with what is going on around the point. Since the only thing about the function that we actually changed was its behavior at $$x = 2$$ this will not change the limit.

Let’s also take a quick look at this function’s graph to see if this says the same thing.

Again, we can see that as we move in towards $$x = 2$$ on our graph the function is still approaching a $$y$$ value of 4. Remember that we are only asking what the function is doing around $$x = 2$$ and we don’t care what the function is actually doing at $$x = 2$$. The graph then also supports the conclusion that the limit is,

$\mathop {\lim }\limits_{x \to 2} g\left( x \right) = 4$

Let’s make the point one more time just to make sure we’ve got it. Limits are not concerned with what is going on at $$x = a$$. Limits are only concerned with what is going on around $$x = a$$. We keep saying this, but it is a very important concept about limits that we must always keep in mind. So, we will take every opportunity to remind ourselves of this idea.

Since limits aren’t concerned with what is actually happening at $$x = a$$ we will, on occasion, see situations like the previous example where the limit at a point and the function value at a point are different. This won’t always happen of course. There are times where the function value and the limit at a point are the same and we will eventually see some examples of those. It is important however, to not get excited about things when the function and the limit do not take the same value at a point. It happens sometimes so we will need to be able to deal with those cases when they arise.

Let’s take a look another example to try and beat this idea into the ground.

Example 3 Estimate the value of the following limit. $\mathop {\lim }\limits_{\theta \to 0} \,\frac{{1 - \cos \left( \theta \right)}}{\theta }$
Show Solution

First don’t get excited about the $$\theta$$ in function. It’s just a letter, just like $$x$$ is a letter! It’s a Greek letter, but it’s a letter and you will be asked to deal with Greek letters on occasion so it’s a good idea to start getting used to them at this point.

Now, also notice that if we plug in $$\theta =0$$ that we will get division by zero and so the function doesn’t exist at this point. Actually, we get 0/0 at this point, but because of the division by zero this function does not exist at $$\theta =0$$.

So, as we did in the first example let’s get a table of values and see what if we can guess what value the function is heading in towards.

$$\theta$$ $$f\left( \theta \right)$$ $$\theta$$ $$f\left( \theta \right)$$
1 0.45969769 -1 -0.45969769
0.1 0.04995835 -0.1 -0.04995835
0.01 0.00499996 -0.01 -0.00499996
0.001 0.00049999 -0.001 -0.00049999

Okay, it looks like the function is moving in towards a value of zero as $$\theta$$ moves in towards 0, from both sides of course.

Therefore, the we will guess that the limit has the value,

$\mathop {\lim }\limits_{\theta \to 0} \,\frac{{1 - \cos \left( \theta \right)}}{\theta } = 0$

So, once again, the limit had a value even though the function didn’t exist at the point we were interested in.

It’s now time to work a couple of more examples that will lead us into the next idea about limits that we’re going to want to discuss.

Example 4 Estimate the value of the following limit. $\mathop {\lim }\limits_{t \to 0} \cos \left( {\frac{\pi }{t}} \right)$
Show Solution

Let’s build up a table of values and see what’s going on with our function in this case.

$$t$$ $$f(t)$$ $$t$$ $$f(t)$$
1 -1 -1 -1
0.1 1 -0.1 1
0.01 1 -0.01 1
0.001 1 -0.001 1

Now, if we were to guess the limit from this table we would guess that the limit is 1. However, if we did make this guess we would be wrong. Consider any of the following function evaluations.

$f\left( {\frac{1}{{2001}}} \right) = - 1\hspace{0.55in}f\left( {\frac{2}{{2001}}} \right) = 0\hspace{0.5in}f\left( {\frac{4}{{4001}}} \right) = \frac{{\sqrt 2 }}{2}$

In all three of these function evaluations we evaluated the function at a number that is less than 0.001 and got three totally different numbers. Recall that the definition of the limit that we’re working with requires that the function be approaching a single value (our guess) as $$t$$ gets closer and closer to the point in question. It doesn’t say that only some of the function values must be getting closer to the guess. It says that all the function values must be getting closer and closer to our guess.

To see what’s happening here a graph of the function would be convenient.

From this graph we can see that as we move in towards $$t = 0$$ the function starts oscillating wildly and in fact the oscillations increases in speed the closer to $$t = 0$$ that we get. Recall from our definition of the limit that in order for a limit to exist the function must be settling down in towards a single value as we get closer to the point in question.

This function clearly does not settle in towards a single number and so this limit does not exist!

This last example points out the drawback of just picking values of the variable and using a table of function values to estimate the value of a limit. The values of the variable that we chose in the previous example were valid and in fact were probably values that many would have picked. In fact, they were exactly the same values we used in the problem before this one and they worked in that problem!

When using a table of values there will always be the possibility that we aren’t choosing the correct values and that we will guess incorrectly for our limit. This is something that we should always keep in mind when doing this to guess the value of limits. In fact, this is such a problem that after this section we will never use a table of values to guess the value of a limit again.

This last example also has shown us that limits do not have to exist. To that point we’ve only seen limits that existed, but that just doesn’t always have to be the case.

Let’s take a look at one more example in this section.

Example 5 Estimate the value of the following limit. $\mathop {\lim }\limits_{t \to 0} H\left( t \right)\hspace{0.25in}{\mbox{where,}}\hspace{0.25in}H\left( t \right) = \left\{ \begin{array}{ll}0 & {\mbox{if }}t < 0\\ 1 & {\mbox{if }}t \ge 0\end{array} \right.$
Show Solution

This function is often called either the Heaviside or step function. We could use a table of values to estimate the limit, but it’s probably just as quick in this case to use the graph so let’s do that. Below is the graph of this function.

We can see from the graph that if we approach $$t = 0$$ from the right side the function is moving in towards a $$y$$ value of 1. Well actually it’s just staying at 1, but in the terminology that we’ve been using in this section it’s moving in towards 1…

Also, if we move in towards $$t = 0$$ from the left the function is moving in towards a $$y$$ value of 0.

According to our definition of the limit the function needs to move in towards a single value as we move in towards $$t = a$$ (from both sides). This isn’t happening in this case and so in this example we will also say that the limit doesn’t exist.

Note that the limit in this example is a little different from the previous example. In the previous example the function did not settle down to a single number as we moved in towards $$t = 0$$. In this example however, the function does settle down to a single number as $$t = 0$$ on either side. The problem is that the number is different on each side of $$t = 0$$. This is an idea that we’ll look at in a little more detail in the next section.

Let’s summarize what we (hopefully) learned in this section. In the first three examples we saw that limits do not care what the function is actually doing at the point in question. They only are concerned with what is happening around the point. In fact, we can have limits at $$x = a$$ even if the function itself does not exist at that point. Likewise, even if a function exists at a point there is no reason (at this point) to think that the limit will have the same value as the function at that point. Sometimes the limit and the function will have the same value at a point and other times they won’t have the same value.

Next, in the third and fourth examples we saw the main reason for not using a table of values to guess the value of a limit. In those examples we used exactly the same set of values, however they only worked in one of the examples. Using tables of values to guess the value of limits is simply not a good way to get the value of a limit. This is the only section in which we will do this. Tables of values should always be your last choice in finding values of limits.

The last two examples showed us that not all limits will in fact exist. We should not get locked into the idea that limits will always exist. In most calculus courses we work with limits that almost always exist and so it’s easy to start thinking that limits always exist. Limits don’t always exist and so don’t get into the habit of assuming that they will.

Finally, we saw in the fourth example that the only way to deal with the limit was to graph the function. Sometimes this is the only way, however this example also illustrated the drawback of using graphs. In order to use a graph to guess the value of the limit you need to be able to actually sketch the graph. For many functions this is not that easy to do.

There is another drawback in using graphs. Even if you have the graph it’s only going to be useful if the $$y$$ value is approaching an integer. If the $$y$$ value is approaching say $$\frac{{ - 15}}{{123}}$$ there is no way that you’re going to be able to guess that value from the graph and we are usually going to want exact values for our limits.

So, while graphs of functions can, on occasion, make your life easier in guessing values of limits they are again probably not the best way to get values of limits. They are only going to be useful if you can get your hands on it and the value of the limit is a “nice” number.

The natural question then is why did we even talk about using tables and/or graphs to estimate limits if they aren’t the best way. There were a couple of reasons.

First, they can help us get a better understanding of what limits are and what they can tell us. If we don’t do at least a couple of limits in this way we might not get all that good of an idea on just what limits are.

The second reason for doing limits in this way is to point out their drawback so that we aren’t tempted to use them all the time!

We will eventually talk about how we really do limits. However, there is one more topic that we need to discuss before doing that. Since this section has already gone on for a while we will talk about this in the next section.

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# Let the sum of the first $n$ terms of a non-constant A.P.,$a_{1},a_{2},a_{3},.....$ be $50n+\frac{n\left ( n-7 \right )}{2}A,$ whrere $A$ is a constant.If $d$ is the common difference of this A.P.,then the ordered pair $\left ( d,a_{50} \right )$ is equal to : Option 1) $\left ( 50,50+46A \right )$ Option 2) $\left ( 50,50+45A \right )$ Option 3)  $\left ( A,50+45A \right )$ Option 4) $\left ( A,50+46A \right )$

Given ,$S_{n}=50n+\frac{n\left ( n-1 \right )}{2}A$...................(1)

$S_{n-1}=50_{\left ( n-1 \right )}+\frac{\left ( n-1 \right )\left ( n-2 \right )}{2}A$.........................(2)

$T_{n}=S_{n}-S_{n-1}=50+A\left ( n-4 \right )$

$T_{1}=50+\left ( -3A \right )$

$T_{2}=50+\left ( -2A \right )$

$d=A$

$T_{50}=50-3A+\left ( 50-1 \right )A$

$=50+46A$

Option 1)

$\left ( 50,50+46A \right )$

Option 2)

$\left ( 50,50+45A \right )$

Option 3)

$\left ( A,50+45A \right )$

Option 4)

$\left ( A,50+46A \right )$

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text:
1

### JEE Main 2018 (Online) 15th April Evening Slot

If |z $-$ 3 + 2i| $\le$ 4 then the difference between the greatest value and the least value of |z| is :
A
$2\sqrt {13}$
B
8
C
4 + $\sqrt {13}$
D
$\sqrt {13}$

## Explanation

$\left| {z - \left( {3 - 2i} \right)} \right| \le 4$ represents a circle whose center is (3, $-$2) and radius = 4.

$\left| z \right|$ = $\left| z -0\right|$  represents the distance of point 'z' from origin (0, 0)

Suppose RS is the normal of the circle passing through origin 'O' and G is its center (3, $-$2).

Here, OR is the least distance

and OS is in the greatest distance

OR = RG $-$ OG and OS = OG + GS . . . . .(1)

As, RG = GS = 4

OG = $\sqrt {{3^2} + \left( { - 2{)^2}} \right)}$ = $\sqrt {9 + 4}$ = $\sqrt {13}$

From (1), OR = 4 $-$ $\sqrt {13}$ and OS = 4 + $\sqrt {13}$

So, required difference = $\left( {4 + \sqrt {13} } \right)$ $-$ $\left( {4 - \sqrt {13} } \right)$

= $\sqrt {13} + \sqrt {13}$ = $2\sqrt {13}$
2

### JEE Main 2018 (Online) 16th April Morning Slot

The least positive integer n for which ${\left( {{{1 + i\sqrt 3 } \over {1 - i\sqrt 3 }}} \right)^n} = 1,$ is :
A
2
B
3
C
5
D
6

## Explanation

${\left( {{{1 + i\sqrt 3 } \over {1 - i\sqrt 3 }}} \right)^n} = 1$

$\Rightarrow \,\,\,\,\,{\left( {{{{{1 + i\sqrt 3 } \over 2}} \over {{{1 - i\sqrt 3 } \over 2}}}} \right)^n} = 1$

We know, $\omega$ = $-$ ${{1 - i\sqrt 3 } \over 2}$

and $\omega$2 = $-$ ${{\left( {1 + i\sqrt 3 } \right)} \over 2}$

$\Rightarrow$ $\,\,\,\,$ ${\left( {{{ - {\omega ^2}} \over { - \omega }}} \right)^n} = 1$

$\Rightarrow$ ($\omega$)n = 1 = $\omega$3

$\Rightarrow$ $\,\,\,\,$ n = 3
3

### JEE Main 2019 (Online) 9th January Morning Slot

Let $\alpha$ and $\beta$ be two roots of the equation x2 + 2x + 2 = 0 , then $\alpha ^{15}$ + $\beta ^{15}$ is equal to :
A
-256
B
512
C
-512
D
256

## Explanation

Given equation,

x2 + 2x + 2 = 0

$\therefore$  x = ${{ - 2 \pm \sqrt {4 - 4.1.2} } \over {2.1}}$

x = $-$ 1 $\pm$ i

$\therefore$  $\alpha$ = $-$ 1 + i

and $\beta$ = $-$ 1 $-$ i

Note :

x + iy = r (cos$\theta$ + isin$\theta$)

$\therefore$  (x + iy)n = rn (cosn$\theta$ + isinn$\theta$)

$\therefore$  $-$ 1 + i = $\sqrt 2$ [cos${{3\pi } \over 4}$ + isin${{3\pi } \over 4}$ ]

$\Rightarrow$  ($-$ 1 + i)15 = ${\left( {\sqrt 2 } \right)^{15}}$ [cos$\left( {{{15.3\pi } \over 4}} \right) + i\sin \left( {{{15.3\pi } \over 4}} \right)$]

And $-$1 $-$ i = $\sqrt 2$ $\left[ {\cos \left( { - {{3\pi } \over 4}} \right) + i\sin \left( { - {{3\pi } \over 4}} \right)} \right]$

= $\sqrt 2 \left[ {\cos {{3\pi } \over 4} - \sin {{3\pi } \over 4}} \right]$

$\therefore$  ($-$1 $-$ i)15 = ${\left( {\sqrt 2 } \right)^{15}}\left[ {\cos \left( {{{15.3\pi } \over 4}} \right) - i\sin \left( {{{15.3\pi } \over 4}} \right)} \right]$

Now

$\alpha$15 + $\beta$15

= ($-$1 + i)15 + ($-$ 1 $-$ i)15

=  ${\left( {\sqrt 2 } \right)^{15}}$ $\left[ {2\cos \left( {{{15.3\pi } \over 4}} \right)} \right]$

= ${\left( {\sqrt 2 } \right)^{15}}\left[ {2\cos \left( {11\pi + {\pi \over 4}} \right)} \right]$

= ${\left( {\sqrt 2 } \right)^{15}}\left[ {2\left( { - \cos {\pi \over 4}} \right)} \right]$

= ${\left( {\sqrt 2 } \right)^{15}} \times 2 \times - {1 \over {\sqrt 2 }}$

= $- {\left( {\sqrt 2 } \right)^{14}}.2$

= $-$ 27 $\times$ 2

= $-$ 28

= $-$ 256
4

### JEE Main 2019 (Online) 9th January Morning Slot

Let
A = $\left\{ {\theta \in \left( { - {\pi \over 2},\pi } \right):{{3 + 2i\sin \theta } \over {1 - 2i\sin \theta }}is\,purely\,imaginary} \right\}$
. Then the sum of the elements in A is :
A
${5\pi \over 6}$
B
$\pi$
C
${3\pi \over 4}$
D
${{2\pi } \over 3}$

## Explanation

Given complex number,

${{3 + 2i\sin \theta } \over {1 - 2i\sin \theta }}$

$= {{\left( {3 + 2i\sin \theta } \right)\left( {1 + 2i\sin \theta } \right)} \over {1 + 4{{\sin }^2}\theta }}$

$= {{3 + 6i\sin \theta + 2i\sin \theta - 4{{\sin }^2}\theta } \over {1 + 4{{\sin }^2}\theta }}$

$= {{\left( {3 - 4{{\sin }^2}\theta } \right) + i\left( {8\sin \theta } \right)} \over {1 + 4{{\sin }^2}\theta }}$

As complex number is purely imaginary, So real part of this complex number is zero.

$\therefore$   ${{3 - 4{{\sin }^2}\theta } \over {1 + 4{{\sin }^2}\theta }}$ = 0

$\Rightarrow$   $3 - 4{\sin ^2}\theta = 0$

$\Rightarrow$   $\sin \theta = \pm {{\sqrt 3 } \over 2}$

as   $\theta$ $\in$ $\left( { - {\pi \over 2},\pi } \right)$

$\therefore$   $\theta$ $=$ $-$ ${\pi \over 3},{\pi \over 3},{{2\pi } \over 3}$

$\therefore$   Sum of those values of A is

$= - {\pi \over 3} + {\pi \over 3} + {{2\pi } \over 3}$

$= {{2\pi } \over 3}$


text:
# Finding expectation value using Heisenberg picture

1. Sep 12, 2005

### broegger

We have a particle in a harmonic oscillator potential. The eigenstates are denoted {|0>,|1>,...,|n>,...}. Initially the particle is in the state |s> = exp(-ipa)|0>, where p is the momentum operator.

I need to find <x> as a function of time using the Heisenberg picture. The problem is, how do I find an expression for |s>? My guess is to express p as p = (some constant)(a- - a+) (where a-/a+ are the ladder operators) and then:

|s> = exp(-i(some constant)(a- - a+)a)|0> = exp(-i(some constant)a)|1>.

But I know this is wrong... What am I supposed to do??

2. Sep 12, 2005

### dextercioby

Well, you've got $|s\rangle$ in the Schrödinger picture. Use the

$$|state\rangle_{Heisenberg}\left(t_{0}\right) = \hat{U}^{\dagger} \left(t,t_{0}\right)|state\rangle_{Schrödinger}(t) \hat{U}\left(t,t_{0}\right)$$

thingy and the expression of the time evolution operator involving the Hamiltonian.

Daniel.

3. Sep 12, 2005

### Physics Monkey

dextercioby,

You can't multiply a ket on both sides with an operator. You are thinking of how operators transform:

$$A_H(t) = U^\dagger(t) A_S U(t)$$

where $$U(t) = e^{-i H t /\hbar}$$

The connection between the Schrodinger state and Heisenberg state is:

$$| \psi \rangle_H = U^\dagger (t) | \psi \rangle_S$$

broegger,

The state you've got there is called a coherent state. You can do the problem using the raising and lowering operators, but your manipulation of the raising and lowering operators is not quite right. You need to be careful with your exponentials. Rememeber that $$e^{A+B} = e^A e^B$$ only if A and B commute. Also I'm not sure how you got just the |1> state in your final step. There is a theorem called the Baker-Campbell-Hausdorff theorem that you might find useful for writing a coherent state in terms of number states.

There is also a more elegant way to do this problem using the Baker-Campbell-Hausdorff theorem (or a version of it) where you need to compute the two objects:

$$e^{i \hat{p}_S a} \hat{p}_S e^{- i \hat{p}_S a}$$
$$e^{i \hat{p}_S a} \hat{x}_S e^{- i \hat{p}_S a}$$

See if you can make progress with these hints.

Last edited: Sep 12, 2005
4. Sep 12, 2005

### reilly

The Heisenberg equations of motion for x are coupled with those for p -- you get them from the commutators, [H,x] and [H,p], and they are directly solvable. The only slightly tricky point is to do everything in harmonic oscillator eigenstates. The integrals involved are standard Gaussian integrals. The rest is standard QM in the Heisenberg Picture as explained in many text books. I would highly recommend doing the problem the old fashioned way, described here, before going on to coherent states, which although very elegant, are not orthogonal and form a so-called over-complete basis.\
Simple is good.
Regards,
Reilly Atkinson

5. Sep 12, 2005

### Physics Monkey

Simple is good. Nevertheless, I think it is possible to mistake familiar methods for simple methods. You can certainly do the whole problem with good ol' position dependent wavefunctions, etc. This is definitely the most familiar method, but I don't think it is most physical approach to the problem. As a physicist I tend to regard the most physical approach as the simplest one. That being said, I think if someone is planning on a future without much quantum mechanics, then the familiar wavefunction approach is fine. On the other hand, if someone is planning to use their QM knowledge again, then I say the quicker you learn the easy physical ways to do problems (even if they involve more complicated mathematics) the better. Physicists would do the problem the way I suggested (for the most part). This could just be the physicist/theorist in my talking though . Any thoughts?

6. Sep 12, 2005

### reilly

Physics Monkey--
I think we are talking matters of taste and style. To me my approach is eminentnly physical -- I was trained roughly 40 years ago, and my profs were steeped in classical mechanics -- so I like x's and p's and wave functions. I'm also a theoretician, but clearly of an older generation -- the simpler the math, the better.

If coherent states work for you in this problem, that's fine. In the early days of QED, after WWII, some prefered Schwinger's approach, many others preferred Feynman's approach.

And, by the way, I think coherent states provide a very powerful tool. But, I think it's best to get the basic groundwork before moving onto more sophisticated topics, including coherent states. To me, the notion of overcompleteness is not an easy one.

I'm happy to state. that both our approaches are equally valid.

Regards,
Reilly Atkinson

7. Sep 13, 2005

### broegger

Thanks everybody. I'm still having problems, though.

I've found the Baker-Hausdorff theorem in my book and I'll try to apply that (I don't have my papers here right now), but I'm also interested in the "wave mechanics" approach. My problem is, I'm bad at working with exponentials of operators and the like.

For example, Physics Monkey mentioned that exp(A+B)=exp(A)exp(B) only if [A,B]=0 and I know that if A|a>=a'|a> (|a> is an eigenket of A with eigenvalue a') then exp(A)|a>=exp(a')|a>. But obviously exp(A+B)|a> = exp(a'+B|a>) is not true. What are the arithmetics of these things??

Update: I found $$\langle x \rangle(t) = a\cos(\omega t)$$ using the Baker-Hausdorff theorem.

Last edited: Sep 13, 2005
8. Sep 13, 2005

### reilly

Actually, there's the Campbell-Baker-Hausdorf thrm (in Optical Coherence &Quantum Optics by Mandell and Wolf, P 519) which says that exp(A+B) = Exp(A)exp(B) if A and B both commute with the commutator of A and B --- [A,[A,B]] = [B,[A,B] = 0. This allows
exp(va* - v*a) = exp(va)exp(v*a), where a and a* are creation and destruction operators -- for the harmonic oscillator/photons a= n(Wq+ip) where W is the oscillator frequency, a* =n(Wq-ip), n is a normalizing constant, and q, and p are respectively position and momentum operators. v is an arbitrary complex constant, v* is its complex conjugate.

So, applied to a vacuum state, a|0>=0;, exp(va*-v*a)|0> = exp(va*)|0> =|v>. the coherent state with with parameter v. Note that <s|v> =/ 0, the coherent states constitute a set of non-orthogonal states. Mandel and Wolf give an excellent account of coherent states, and their use in quantum optics.(Note, [a,a*]=1, so both a and a* commute with their commutator. Their book, however, presumes familiarity with the basics of QM.

I still suggest that the approach I outlined is straighforward, and certainly tedious. You'll find that both x and p are periodic.

Regards,
Reilly Atkinson

Last edited: Sep 13, 2005
9. Sep 13, 2005

### Physics Monkey

reilly,

Let me first say your points are all well taken. Certainly if the wavefunction approach seems more physical to you, then I won't argue. It is a matter of taste and style.

Secondly, I don't have my copy of Mandel and Wolf handy, but if you copied the theorem correctly then the book has a misprint. The correct statement of the Baker-Cambpell-Hausdorff theorem in the case when [A,[A,B]] = [B,[A,B]] = 0 is,
$$e^{A+B} = e^A e^B e^{-1/2 \,[A,B]}$$

This means that the so called 'displacement' operator is,
$$D(v) = e^{v a^\dagger - v^* a} = e^{v a^\dagger} e^{-v^* a} e^{-1/2\, v^* v}$$
This is important because the coherent state is really given by,
$$| v \rangle = e^{-1/2\, v^* v} e^{v a^\dagger} | 0 \rangle$$

broegger,

I'm glad to hear that you figured out the problem. Are you still interested in using the wavefunction approach as well?

10. Sep 13, 2005

### reilly

Physics Monkey -- You are correct. From time-to-time I blow it, and I just did. I have a bad habit of forgetting about normalizing factors. Thanks

Broegger:
I think the wave function approach goes something like this:
<p|s> = const*exp(-ipa - cnst p*p), and @ t=o, x= id/dp. Get the time stuff, and then the expectation value is a gaussian integral. I think your answer is not correct, if only because no oscillator parameters are involved.

Regards,
Reilly Atkinson

Last edited: Sep 13, 2005
11. Sep 14, 2005

### broegger

My answer, obtained by the Baker-Hausdorff theorem, was correct, it has just been confirmed :)

Thanks, guys.

Physicsmonkey: Is the wave mechanics approach related to the fact that exp(-ipa/h) is the translation-in-space operator?

12. Sep 14, 2005

### Physics Monkey

The wavefunction approach really just means that you use the position basis (or just as well, the momentum basis) to calculate expectation values. The fact that your state is defined in terms of the translation-in-space operator and the ground state means that the wavefunction is easy to obtain: the wavefunction $$\psi_s(x) =\langle x | s \rangle$$ of the coherent state is just the ground state wavefunction displaced by an amount a. Using this fact and the formulas you derived for the time dependence of the Heisenberg operators in terms of the Schrodinger picture operators, you can calculate the expectation value by doing two integrals.

13. Sep 15, 2005

### broegger

Thanks. I must say I like the Baker-Hausdorff-way better, although the wave mechanics approach is more straightforward (and probably what we were supposed to do).

14. Sep 19, 2005

### dextercioby

I must have had a very bad day. :(( Sorry, it won't happen again.

Daniel.


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# Primitive Elements in Free Groups

Martin J. Evans
Proceedings of the American Mathematical Society
Vol. 106, No. 2 (Jun., 1989), pp. 313-316
DOI: 10.2307/2048805
Stable URL: http://www.jstor.org/stable/2048805
Page Count: 4
Let Fn denote the free group of rank n generated by x1, x2,..., xn. We say that y ∈ Fn is a primitive element of Fn if it is contained in a set of free generators of Fn. In this note we construct, for each integer n ≥ 4, an (n - 1)-generator group H that has an n-generator, 2-relator presentation $H = \langlex_1,\ldots, x_n\midr_1, r_2\rangle$ such that the normal closure of $\{r_1, r_2 \}$ in Fn does not contain a primitive element of Fn.


text:
### Re: Switch user command not working user Net:SSH2 module

by salva (Abbot)
 on Oct 29, 2013 at 09:21 UTC ( #1060127=note: print w/replies, xml ) Need Help??

Net::SSH2 runs every command in a different shell session. Read the FAQ entry "Can't change working directory" from the Net::OpenSSH docs for a more detailed explanation.

There are two ways to do what you want:

One is to start a shell on the remote shell and then simulate an interactive dialogue with it (usually, you use Expect for that, though Expect doesn't work on Windows*). This is hard and error prone, as you will have to look into the command output for the prompt in order to detect when some command is done.

The other way is to combine the sequence of commands into a single one. But, su usually asks for the password and so you will have to send it through its tty.

su root -c "whoami"
[download]

An easier way is to use sudo, that can be configured to not ask for a password, or alternatively, in recent versions, to get the passwords from stdin. For instance, using Net::SSH::Any:

my $ssh2 = Net::SSH::Any->new('192.168.XX.XXX, user => 'root', passwor +d => 'root'); print$ssh2->capture({stdin_data => "$sudo_passwd\n"}, 'sudo', '-Sk', '-U',$user,
'-p', '',
'--',
'whoami');
[download]

*) unless you use the perl from Cygwin.

Create A New User
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node history
Node Type: note [id://1060127]
help
Chatterbox?
 Corion joins another telephone call :-/ [Discipulus]: also The Life and Opinions of the Tomcat Murr [Discipulus]: I dunno if i can read Dumas in french.. must try some chapter to see erix . o O( "what fools the french are, Jeeves" ) [choroba]: Also some Достое́вс кий [Discipulus]: mmh windows understand something like: cd c:\\\\path\\\\to weird..

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# Why velocity increases when radius decreases

1. Sep 15, 2007

### vivinisaac

pls explain what is angular momentum
and if possible explain why velocity increases when radius decreases (not mathematically)

2. Sep 15, 2007

### rcgldr

Linear momentum is equal to speed x mass.

Angular momentum is basically the same, except the mass is traveling in a circular path, it's relative to the angular velocity (speed of rotation) x mass x radius. The speed of the mass is equal to the angular velocity x radius, and if mass and angular momentum are constant, than a decrease in radius requires an increase in angular velocity, while the speed of the mass remains constant (same speed in a smaller circler means a faster rate of rotation).

3. Sep 16, 2007

### Loren Booda

Also, angular momentum is a conserved quantity, having the units of gm-cm2/sec (or erg-sec) in the cgs system. Angular momentum is the cross product of a momentum vector and its radial vector, thus oriented perpendicular to both.


text:
# How to Call a function from another script?

I just learned in a C# tutorial that we can declare var name = new ClassNameOfOtherScript(); to create a new class object, then use the object to call the function in that class, but it doesn't seem to be working in Unity.

Am I understanding it wrong or does it work differently in Unity?

I have a class Cube1 this is attached to an object with SpriteRenderer enabled. Then I have another script ClockController which is not attach to any object. When I run it, the color of cube does not change to red?

public class Cube1 : MonoBehaviour
{

public void Rectangle()
{
GetComponent<SpriteRenderer>().color = Color.red;
}

}

public class ClockController : MonoBehaviour
{

void Start()
{
var cube1 = new Cube1();
cube1.Rectangle();
}
}

• Why would the colour of your existing Cube1 object change when you called the colour change function on a new Cube1 object? (ie. NOT the one that already exists in your scene/project) Did you try searching for past answers about how to get a reference to an existing object to call a method on it? Jan 11 '21 at 13:03
• Tested both code and noticed the different, indeed putting the title getting a reference on(existing) object would be more appropriate. Been learning C# fundamental tutorial for sometime yet I do not realize I'm on the wrong direction regarding the usage of "new" operator. Thank again!! Jan 11 '21 at 15:17

Well you are doing var cube1 = new Cube1(); which is how you would make an object in C#, but since it is Unity, and this is a MonoBehaviour, you are not supposed to make it like that.

Instead, make a new GameObject and attach the Cube1 script to it.

void Start()
{
// make a new gameobject.
GameObject gameObject = new GameObject();

// adding a SpriteRenderer as your Cube1 will try to access it.
// -- please also see the bottom note of the answer. --

// attach a Cube1 script to it and store the reference to it in a var.

// call the function on that cube.
cube.Rectangle();
}

Sidenote, if you don't make an object in the scene that holds the ClockController, it won't execute the Start() method.

When you already have an existing GameObject (instead of making a new one) that has a script you want to access, you can use gameObject.GetComponent<Cube1>() , store it in a variable (if necessary) and access it that way. You just need to get that specific GameObject to begin with.

Extra info on component dependencies:

As pointed out by @DMGregory, cube.Rectangle() will try to find/access a SpriteRenderer on the cube object, which would have to be added first. You could add [RequireComponent(typeof(SpriteRenderer))] which means that whenever you add a Cube1 script to a GameObject, the required component is added aswell if it is not already there.

I'd suggest to take a look at the docs: https://docs.unity3d.com/ScriptReference/RequireComponent.html

Alternatives would probably be making sure in your Cube1 script that at some point a SpriteRenderer is attached, but that's all going off-topic I think.

• Note that since Cube1.Rectangle() tries to get a reference to an attached SpriteRenderer component, it does not make sense to call this on a new object that has no SpriteRenderer on it, as shown in this answer. The as Cube1 is also unnecessary — the return type is already Cube1 Jan 11 '21 at 13:23
• Ah thanks, I was not sure about the as Cube1 part so looked up the docs (and saw it there). Will update my answer. Sharp one on the SpriteRenderer, guess I was too focussed on only answering how to call the function. Jan 11 '21 at 13:39
• Note that the as is shown in the docs only for the overloads that take a type or string as an argument. It is not present in the example for the generic version that you're using here. Jan 11 '21 at 13:42