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Thalassemia describes a group of inherited disorders characterized by reduced or absent amounts of hemoglobin, the oxygen-carrying protein inside the red blood cells. There are two basic groups of thalassemia disorders: alpha thalassemia and beta thalassemia. These conditions cause varying degrees of anemia, which can range from insignificant to life threatening.
All types of thalassemias are considered quantitative diseases of hemoglobin, because the quantity of hemoglobin produced is reduced or absent. Usual adult hemoglobin is made up of three components: alpha globin, beta globin, and heme. Thalassemias are classified according to the globin that is affected, hence the names alpha and beta thalassemia. Although both classes of thalassemia affect the same protein, the alpha and beta thalassemias are distinct diseases that affect the body in different ways.
Beta thalassemia may be the most best-known type of thalassemia and is also called Cooley's anemia. It is caused by a change in the gene for the beta globin component of hemoglobin. Beta thalassemia causes variable anemia that can range from moderate to severe, depending in part on the exact genetic change underlying the disease. Beta thalassemia can be classified based on clinical symptoms. Beta thalassemia major usually causes severe anemia that can occur within months after birth. If left untreated, severe anemia can result in insufficient growth and development, as well as other common physical complications that can lead to a dramatically decreased life-expectancy. Fortunately, in developed countries beta thalassemia is usually identified by screening in the newborn period, before symptoms have developed. Children who are identified early can be started on ongoing blood transfusion therapy as needed. Although transfusion therapy prevents many of the complications of severe anemia, the body is unable to eliminate the excess iron contained in the transfused blood. Over time, the excess iron deposits in tissues and organs, resulting in damage and organ failure. Another medication must be administered to help the body eliminate the excess iron and prevent iron-over-load complications. Beta thalassemia intermedia describes the disease in individuals who have moderate anemia that only requires blood transfusions intermittently, if at all.
Alpha thalassemia is the result of changes in the genes for the alpha globin component of hemoglobin. There are two main types of alpha thalassemia disease: hemoglobin H disease and alpha thalassemia major. The two diseases are quite different from beta thalassemia as well as from one another. Individuals with hemoglobin H disease can experience events of hemolytic anemia—anemia caused by the rapid breakdown of the red blood cells. These events are thought to be triggered by various environmental causes, such as infection and/or exposure to certain chemicals. Hemoglobin H disease is in most cases milder than beta thalassemia. It does not generally require transfusion therapy. Alpha thalassemia major is a very serious disease that results in severe anemia that begins even before birth. Most affected babies do not survive to be born or die shortly after birth.
The thalassemias are among the most common genetic diseases worldwide. Both alpha and beta thalassemia have been described in individuals of almost every ancestry, but the conditions are more common among certain ethnic groups. Unaffected carriers of all types of thalassemia traits do not experience health problems. In fact, the thalassemia trait is protective against malaria, a disease caused by blood-borne parasites transmitted through mosquito bites. According to a widely accepted theory, most genetic changes—mutations—that cause thalassemia occurred multiple generations ago. Coincidentally, these mutations increased the likelihood that carriers would survive malaria infection. Survivors passed the mutation onto their offspring, and the trait became established throughout areas where malaria is common. As populations migrated, so did the thalassemia traits.
Beta thalassemia trait is seen most commonly in people with the following ancestry: Mediterranean (including North African, and particularly Italian and Greek), Middle Eastern, Indian, African, Chinese, and Southeast Asian (including Vietnamese, Laotian, Thai, Singaporean, Filipino, Cambodian, Malaysian, Burmese, and Indonesian). Alpha-thalassemia trait is seen with increased frequency in the same ethnic groups. However, there are different types of alpha thalassemia traits within these populations. The frequency of hemoglobin H disease and alpha thalassemia major depends on the type of alpha thalassemia trait. The populations in which alpha thalassemia diseases are most common include Southeast Asians and Chinese (particularly Southern Chinese).
It is difficult to obtain accurate prevalence figures for various types of thalassemia within different populations. This difficulty arises due to testing limitations in determining exact genetic diagnoses, as well as the fact that many studies have focused on small, biased hospital populations.
Two studies reflect prevalence figures that can be helpful counseling families and determining who to screen for beta thalassemia. Between the years of 1990 and 1996, the State of California screened more than 3.1 million infants born in the state for beta thalassemia. Approximately 1 in 114,000 infants had beta thalassemia major, with prevalence rates being highest among Asian Indians (about one in 4,000), Southeast Asians (about one in 10,000), and Middle Easterners (about one in 7,000). Another type of beta thalassemia disease, E/beta thalassemia, was represented in approximately one in 110,000 births, all of which occurred in families of Southeast Asian ancestry. Among Southeast Asians, the prevalence of E/beta thalassemia was approximately one in 2,600 births. This is in keeping with the observation that hemoglobin E trait carrier rates are relatively high within the Southeast Asian population: 16% in a study of 768 immigrants to California, and up to 25% in some specific Southeast Asian populations such as Cambodians. While these California studies address some of the limitations of earlier population studies, the pattern observed in California is expected to be different in other areas of the United States and the world. For example, Italians are underrepresented in this population when compared to the population of the East Coast of the United States.
Determining prevalence figures for alpha thalassemia is even more difficult due to increased limitations in diagnostic testing. All types of alpha thalassemia disease are most common among people of Southeast Asian and Chinese descent, for reasons that become clearer with an understanding of the underlying genetics of alpha thalassemia. One study of 500 pregnant women in Northern Thailand estimated a frequency of one in 500 pregnancies affected by alpha thalassemia major, for example. Prevalence of alpha thalassemia disease is significantly lower in the United States primarily because of immigration patterns; although at least one state, California, has observed growing hemoglobin H disease incidence rates that are high enough to justify universal newborn screening for the condition.
Humans normally make several types of the oxygen-carrying protein hemoglobin. An individual's stage in development determines whether he or she makes primarily embryonic, fetal, or adult hemoglobins. All types of hemoglobin are made of three components: heme, alpha (or alpha-like) globin, and beta (or beta-like) globin. All types of thalassemia are caused by changes in either the alpha- or beta-globin gene. These changes cause little or no globin to be produced. The thalassemias are, therefore, considered quantitative hemoglobin diseases. All types of thalassemias are recessively inherited, meaning that a genetic change must be inherited from both the mother and the father. The severity of the disease is influenced by the exact thalassemia mutations inherited, as well as other genetic and environmental factors. There are rare exceptions, notably with beta thalassemia, where globin gene mutations exhibit a dominant pattern of inheritance in which only one gene needs to be altered in order to see disease expression. Scientists continue to study the causes. For instance, a new mutation for alpha-thalassemia was discovered for the first time among Iranian patients in 2004.
BETA-THALASSEMIA. Most individuals have two normal copies of the beta globin gene, which is located on chromosome 11 and makes the beta globin component of normal adult hemoglobin, hemoglobin A. There are approximately 100 genetic mutations that have been described that cause beta thalassemia, designated as either beta0 or beta + mutations. No beta globin is produced with a beta0 mutation, and only a small fraction of the normal amount of beta globin is produced with a beta + mutation.
When an individual has one normal beta globin gene and one with a beta thalassemia mutation, he or she is said to carry the beta thalassemia trait. Beta thalassemia trait, like other hemoglobin traits, is protective against malaria infection. Trait status is generally thought not to cause health problems, although some women with beta thalassemia trait may have an increased tendency toward anemia during pregnancy.
When two members of a couple carry the beta thalassemia trait, there is a 25% chance that each of their children will inherit beta thalassemia disease by inheriting two beta thalassemia mutations, one from each parent. The clinical severity of the beta thalassemia disease—whether an individual has beta thalassemia intermedia or beta thalassemia major—will depend largely on whether the mutations inherited are beta0 thalassemia or beta + thalassemia mutations. Two beta0 mutations generally lead to beta thalassemia major, and two beta+ thalassemia mutations generally lead to beta thalassemia intermedia. Inheritance of one beta0 and one beta + thalassemia mutation tends to be less predictable.
Although relatively uncommon, there are other thalassemia-like mutations that can affect the beta globin gene. Hemoglobin E is the result of a substitution of a single nucleotide. This change results in a structurally altered hemoglobin that is produced in decreased amounts. Therefore, hemoglobin E is unique in that it is both a quantitative (i.e. thalassemia-like) and qualitative trait. When co-inherited with a beta thalassemia trait, it causes a disease that is almost indistinguishable from beta thalassemia disease. Large deletions around and including the beta globin gene can lead to delta/beta thalassemia or hereditary persistence of fetal hemoglobin (HPFH). Interestingly, delta/beta thalassemia trait behaves very similarly to beta thalassemia trait in its clinical manifestations. However, HPFH trait does not tend to cause hemoglobin disease when co-inherited with a second thalassemia or other beta globin mutation.
ALPHA-THALASSEMIA. Most individuals have four normal copies of the alpha globin gene, two copies on each chromosome 16. These genes make the alpha globin component of normal adult hemoglobin, which is called hemoglobin A. Alpha globin is also a component of fetal hemoglobin and the other major adult hemoglobin called hemoglobin A2. Mutations of the alpha globin genes are usually deletions of the gene, resulting in absent production of alpha globin. Since there are four genes (instead of the usual two) to consider when looking at alpha globin gene inheritance, there are several alpha globin types that are possible.
Absence of one alpha globin gene leads to a condition known as silent alpha thalassemia trait. This condition causes no health problems and can be detected only by special genetic testing. Alpha thalassemia trait occurs when two alpha globin genes are missing. This can occur in two ways. The genes may be deleted from the same chromosome, causing the 'cis' type of alpha thalassemia trait. Alternately, they may be deleted from different chromosomes, causing the 'trans' type of alpha thalassemia trait. In both instances, there are no associated health problems, although the trait status may be detected by more routine blood screening.
Hemoglobin H disease results from the deletion of three alpha globin genes, such that there is only one functioning gene. Typically, this can occur when one parent carries the silent alpha thalassemia trait, and the other parent carries the 'cis' type of the alpha thalassemia trait. In this situation, there is a 25% chance for hemoglobin H disease in each of such a couple's children.
Hemoglobin H disease-like symptoms can also be a part of a unique condition called alpha thalassemia mental retardation syndrome. Alpha thalassemia mental retardation syndrome can be caused by a deletion of a significant amount of chromosome 16, affecting the alpha globin genes. This is usually not inherited, but rather occurs sporadically in the affected individual. Affected individuals have mild hemoglobin H disease, mild-to-moderate mental retardation, and characteristic facial features. This syndrome can also occur as a sex-linked form in which a mutation is inherited in a particular gene on the X-chromosome. This gene influences alpha globin production, as well as various other developmental processes. Individuals affected with this form of the syndrome tend to have more severe mental retardation, delayed development, nearly absent speech, characteristic facial features, and genital-urinary abnormalities. The remaining discussion will focus only on aspects of hemoglobin H disease.
Alpha thalassemia major results from the deletion of all four alpha globin genes, such that there are no functioning alpha globin genes. This can occur when both parents carry the 'cis' type of the alpha thalassemia trait. In this situation, there is a 25% chance for alpha thalassemia major in each of such a couple's children.
Beta thalassemia major is characterized by severe anemia that can begin months after birth. In the United States and other developed countries beta thalassemia is identified and treated early and effectively. Therefore, the following discussion of symptoms applies primarily to affected individuals in the past and unfortunately in some underdeveloped countries now. If untreated, beta thalassemia major can lead to severe lethargy, paleness, and delays in growth and development. The body attempts to compensate by producing more blood, which is made inside the bones in the marrow. However, this is ineffective without the needed genetic instructions to make enough functioning hemoglobin. Instead, obvious bone expansion and changes occur that cause characteristic facial and other changes in appearance, as well as increased risk of fractures. Severe anemia taxes other organs in the body—such as the heart, spleen, and liver—which must work harder than usual. This can lead to heart failure, as well as enlargement and other problems of the liver and spleen. When untreated, beta thalassemia major generally results in childhood death, usually due to heart failure. In 2004, the first known heart attack associated with beta thalassemia major was reported. Fortunately, in developed countries diagnosis is usually made early, often before symptoms have begun. This allows for treatment with blood transfusion therapy, which can prevent most of the complications of the severe anemia caused by beta thalassemia major. Individuals with beta thalassemia intermedia have a more moderate anemia that may only require treatment with transfusion intermittently, such as when infections occur and stress the body. As a person with beta thalassemia intermedia gets older, however, the need for blood transfusions may increase to the point that they are required on a regular basis. When this occurs their disease becomes more similar to beta thalassemia major. Other genetic and environmental factors can influence the course of the disease as well. For example, co-inheritance of one or two alpha thalassemia mutations can tend to ameliorate some of the symptoms of beta thalassemia disease, which result in part from an imbalance in the amount of alpha- and beta-globin present in the red blood cells.
Hemoglobin h disease
Absence of three alpha globin genes causes an imbalance of alpha and beta globin proteins in the red blood cells. The excess beta globin proteins tend to come together to form hemoglobin H, which is unable to release oxygen to the tissues. In addition, hemoglobin H tends to precipitate out in the cells, causing damage to the red blood cell membrane. When affected individuals are exposed to certain drugs and chemicals known to make the membrane more fragile, the cells are thought to become vulnerable to breakdown in large numbers, a complication called hemolytic anemia. Fever and infection are also considered to be triggers of hemolytic anemia in hemoglobin H disease. This can result in fatigue, paleness, and a yellow discoloration of the skin and whites of eyes called jaundice. Usually, the anemia is mild enough not to require treatment. Severe anemia events may require blood transfusion, however, and are usually accompanied by such other symptoms as dark feces or urine and abdominal or back pain. These events are uncommon in hemoglobin H disease, although they occur more frequently in a more serious type of hemoglobin H disease called hemoglobin H/Constant Spring disease. Individuals effected with this type of hemoglobin H disease are also more likely to have enlargement of and other problems with the spleen.
Alpha thalassemia major
Because alpha globin is a necessary component of all major hemoglobins and some minor hemoglobins, absence of all functioning alpha globin genes leads to serious medical consequences that begin even before birth. Affected fetuses develop severe anemia as early as the first trimester of pregnancy. The placenta, heart, liver, spleen, and adrenal glands may all become enlarged. Fluid can begin collecting throughout the body as early as the start of the second trimester, causing damage to developing tissues and organs. Growth retardation is also common. Affected fetuses usually miscarry or die shortly after birth. In addition, women carrying affected fetuses are at increased risk of developing complications of pregnancy and delivery. Up to 80% of such women develop toxemia, a disturbance of metabolism that can potentially lead to convulsions and coma. Other maternal complications include premature delivery and increased rates of delivery by cesarean section, as well as hemorrhage after delivery.
Thalassemia may be suspected if an individual shows signs that are suggestive of the disease. In all cases, however, laboratory diagnosis is essential to confirm the exact diagnosis and to allow for the provision of accurate genetic counseling about recurrence risks and testing options for parents and affected individuals. Screening is likewise recommended to determine trait status for individuals of high-risk ethnic groups.
The following tests are used to screen for thalassemia disease and/or trait:
hemoglobin electrophoresis with quantitative hemoglobin A2 and hemoglobin F
free erythrocyte-protoporphyrin (or ferritin or other studies of serum iron levels)
A complete blood count will identify low levels of hemoglobin, small red blood cells, and other red blood cell abnormalities that are characteristic of a thalassemia diagnosis. Since thalassemia trait can sometimes be difficult to distinguish from iron deficiency, tests to evaluate iron levels are important. A hemoglobin electrophoresis is a test that can help identify the types and quantities of hemoglobin made by an individual. This test uses an electric field applied across a slab of gel-like material. Hemoglobins migrate through this gel at various rates and to specific locations, depending on their size, shape, and electrical charge. Isoelectric focusing and high-performance liquid chromatography (HPLC) use similar principles to separate hemoglobins and can be used instead of or in various combinations with hemoglobin electrophoresis to determine the types and quantities of hemoglobin present. Hemoglobin electrophoresis results are usually within the normal range for all types of alpha thalassemia. However, hemoglobin A2 levels and sometimes hemoglobin F levels are elevated when beta thalassemia disease or trait is present. Hemoglobin electrophoresis can also detect structurally abnormal hemoglobins that may be co-inherited with a thalassemia trait to cause thalassemia disease (i.e., hemoglobin E) or other types of hemoglobin disease (i.e., sickle hemoglobin). Sometimes DNA testing is needed in addition to the above screening tests. This can be performed to help confirm the diagnosis and establish the exact genetic type of thalassemia.
Diagnosis of thalassemia can occur under various circumstances and at various ages. Several states offer thalassemia screening as part of the usual battery of blood tests done for newborns. This allows for early identification and treatment. Thalassemia can be identified before birth through the use of prenatal diagnosis. Chorionic villus sampling (CVS) can be offered as early as 10 weeks of pregnancy and involves removing a sample of the placenta made by the baby and testing the cells. CVS carries a risk of causing a miscarriage that is between 0.5%-1%. Amniocentesis is generally offered between 15 and 22 weeks of pregnancy, but can sometimes be offered earlier. Two to three tablespoons of the fluid surrounding the baby is removed. This fluid contains fetal cells that can be tested. The risk of miscarriage associated with amniocentesis ranges from 0.33-0.5%. Pregnant woman and couples may choose prenatal testing in order to prepare for the birth of a baby that may have thalassemia. Alternately, knowing the diagnosis during pregnancy allows for the option of pregnancy termination. Preimplantation genetic diagnosis (PGD) is a relatively new technique that involves in-vitro fertilization followed by genetic testing of one cell from each developing embryo. Only the embryos unaffected by sickle cell disease are transferred back into the uterus. PGD is currently available on a research basis only and is relatively expensive.
Individuals with beta thalassemia major receive regular blood transfusions, usually on a monthly basis. This helps prevent severe anemia and allows for more normal growth and development. Transfusion therapy does have limitations, however. Individuals can develop reactions to certain proteins in the blood—called a transfusion reaction. This can make locating appropriately matched donor blood more difficult. Although blood supplies in the United States are very safe, particularly relative to the past and to other areas of the world, there remains an increased risk of exposure to such blood-borne infections as hepatitis. Additionally, the body is not able to get rid of the excess iron that accompanies each transfusion. An additional medication called desferoxamine is administered, usually five nights per week over a period of several hours, using an automatic pump that can be used during sleep or taken anywhere the person goes. This medication is able to bind to the excess iron, which can then be eliminated through urine. If desferoxamine is not used regularly or is unavailable, iron overload can develop and cause tissue damage and organ damage and failure. The heart, liver, and endocrine organs are particularly vulnerable. Desferoxamine itself may rarely produce allergic or toxic side effects, including hearing damage. Signs of desferoxamine toxicity are screened for and generally develop in individuals who overuse the medication when body iron levels are sufficiently low. Overall, however, transfusion and desferoxamine therapy have increased the life expectancy of individuals with the most severe types of beta thalassemia major to the 4th or 5th decade. This can be expected to improve with time and increased developments in treatment, as well as for those with more mild forms of the disease.
New treatments offer additional options for some individuals with beta thalassemia major. There are various medications that target the production of red blood cells (i.e. erythropoeitin) or fetal hemoglobin (i.e. hydroxyurea and butyrate). Their effectiveness in ameliorating the severity of beta thalassemia is currently being investigated. Another promising new treatment is bone marrow transplantation, in which the bone marrow of an affected individual is replaced with the bone marrow of an unaffected donor. If successful, this treatment can provide a cure. However, there is an approximately 10-15% chance the procedure could be unsuccessful (i.e. the thalassemia returns); result in complications (i.e. graft-versus-host disease); or result in death. The risk for specific individuals depends on current health status, age, and other factors. Because of the risks involved and the fact that beta thalassemia is a treatable condition, transplant physicians require a brother or sister donor who has an identically matched tissue type, called HLA type. HLA type refers to the unique set of proteins present on each individual's cells, which allows the immune system to recognize "self" from "foreign." HLA type is genetically determined, so there is a 25% chance for two siblings to be a match. Transplant physicians and researchers are also investigating ways to improve the safety and effectiveness of bone marrow transplantation. Using newborn sibling umbilical cord blood—the blood from the placenta that is otherwise discarded after birth but contains cells that can go on to make bone marrow—seems to provide a safer and perhaps more effective source of donor cells. Donors and recipients may not have to be perfect HLA matches for a successful transplant using cord blood cells. Trials are also underway to determine the effectiveness of "partial transplants," in which a safer transplant procedure is used to replace only a percentage of the affected individual's bone marrow. Other possible treatments on the horizon may include gene therapy techniques aimed at increasing the amount of normal hemoglobin the body is able to make.
Hemoglobin H disease is a relatively mild form of thalassemia that may go unrecognized. It is not generally considered a condition that will reduce one's life expectancy. Education is an important part of managing the health of an individual with hemoglobin H disease. It is important to be able to recognize the signs of severe anemia that require medical attention. It is also important to be aware of the medications, chemicals, and other exposures to avoid due to the theoretical risk they pose of causing a severe anemia event. When severe anemia occurs, it is treated with blood transfusion therapy. For individuals with hemoglobin H disease, this is rarely required. For those with the hemoglobin H/Constant Spring form of the disease, the need for transfusions may be intermittent or ongoing, perhaps on a monthly basis and requiring desferoxamine treatment. Individuals with this more severe form of the disease may also have an increased chance of requiring removal of an enlarged and/or overactive spleen.
Anemia — A blood condition in which the level of hemoglobin or the number of red blood cells falls below normal values. Common symptoms include paleness, fatigue, and shortness of breath.
Bilirubin — A yellow pigment that is the end result of hemoglobin breakdown. This pigment is metabolized in the liver and excreted from the body through the bile. Bloodstream levels are normally low; however, extensive red cell destruction leads to excessive bilirubin formation and jaundice.
Bone marrow — A spongy tissue located in the hollow centers of certain bones, such as the skull and hip bones. Bone marrow is the site of blood cell generation.
Bone marrow transplantation — A medical procedure used to treat some diseases that arise from defective blood cell formation in the bone marrow. Healthy bone marrow is extracted from a donor to replace the marrow in an ailing individual. Proteins on the surface of bone marrow cells must be identical or very closely matched between a donor and the recipient.
Desferoxamine — The primary drug used in iron chelation therapy. It aids in counteracting the life-threatening buildup of iron in the body associated with long-term blood transfusions.
Globin — One of the component protein molecules found in hemoglobin. Normal adult hemoglobin has a pair each of alpha-globin and beta-globin molecules.
Heme — The iron-containing molecule in hemoglobin that serves as the site for oxygen binding.
Hemoglobin — Protein-iron compound in the blood that carries oxygen to the cells and carries carbon dioxide away from the cells.
Hemoglobin A — Normal adult hemoglobin that contains a heme molecule, two alpha-globin molecules, and two beta-globin molecules.
Hemoglobin electrophoresis — A laboratory test that separates molecules based on their size, shape, or electrical charge.
Hepatomegaly — An abnormally large liver.
HLA type — Refers to the unique set of proteins called human leukocyte antigens. These proteins are present on each individual's cell and allow the immune system to recognize 'self' from 'foreign'. HLA type is particularly important in organ and tissue transplantation.
Hydroxyurea — A drug that has been shown to induce production of fetal hemoglobin. Fetal hemoglobin has a pair of gamma-globin molecules in place of the typical beta-globins of adult hemoglobin. Higher-than-normal levels of fetal hemoglobin can ameliorate some of the symptoms of thalassemia.
Iron overload — A side effect of frequent blood transfusions in which the body accumulates abnormally high levels of iron. Iron deposits can form in organs, particularly the heart, and cause life-threatening damage.
Jaundice — Yellowing of the skin or eyes due to excess of bilirubin in the blood.
Mutation — A permanent change in the genetic material that may alter a trait or characteristic of an individual, or manifest as disease, and can be transmitted to offspring.
Placenta — The organ responsible for oxygen and nutrition exchange between a pregnant mother and her developing baby.
Red blood cell — Hemoglobin-containing blood cells that transport oxygen from the lungs to tissues. In the tissues, the red blood cells exchange their oxygen for carbon dioxide, which is brought back to the lungs to be exhaled.
Screening — Process through which carriers of a trait may be identified within a population.
Splenomegaly — Enlargement of the spleen.
Because alpha thalassemia major is most often a condition that is fatal in the prenatal or newborn period, treatment has previously been focused on identifying affected pregnancies in order to provide appropriate management to reduce potential maternal complications. Pregnancy termination provides one form of management. Increased prenatal surveillance and early treatment of maternal complications is an approach that is appropriate for mothers who wish to continue their pregnancy with the knowledge that the baby will most likely not survive. In recent years, there have been a handful of infants with this condition who have survived long-term. Most of these infants received experimental treatment including transfusions before birth, early delivery, and even bone marrow transplantation before birth, although the latter procedure has not yet been successful. For those infants that survive to delivery, there seems to be an increased risk of developmental problems and physical effects, particularly heart and genital malformations. Otherwise, their medical outlook is similar to a child with beta thalassemia major, with the important exception that ongoing, life-long blood transfusions begin right at birth.
As discussed above, the prognosis for individuals with the most serious types of thalassemia has improved drastically in the last several years following recent medical advances in transfusion, chemo-, and transplantation therapy. Advances continue and promise to improve the life expectancy and quality of life further for affected individuals.
"First Known Heart Attack Associated With Beta-thalassemia Major Reported." Heart Disease Weekly February 22, 2004: 10.
"Novel Alpha-thalassemia Mutations Identified." Hematology Week January 26, 2004: 19.
Children's Blood Foundation. 333 East 38th St., Room 830, New York, NY 10016-2745. (212) 297-4336. cfg@nyh.med.cornell.edu.
Cooley's Anemia Foundation, Inc. 129-09 26th Ave. #203, Flushing, NY 11354. (800) 522-7222 or (718) 321-2873. http://www.thalassemia.org.
March of Dimes Birth Defects Foundation. 1275 Mamaroneck Ave., White Plains, NY 10605. (888) 663-4637. resourcecenter@modimes.org. http://www.modimes.org.
National Heart, Lung, and Blood Institute. PO Box 30105, Bethseda, MD 20824-0105. (301) 592-8573. nhlbiinfo@rover.nhlbi.nih.gov. http://www.nhlbi.nih.gov.
National Organization for Rare Disorders (NORD). PO Box 8923, New Fairfield, CT 06812-8923. (203) 746-6518 or (800) 999-6673. Fax: (203) 746-6481. http://www.rarediseases.org.
Bojanowski J. "Alpha Thalassemia Major: The Possibility of Long-Term Survival." Pamphlet from the Northern California Comprehensive Thalassemia Center. (1999).
Children's Hospital Oakland, Northern California Comprehensive Thalassemia Center website. http://www.thalassemia.com.
Cooley's Anemia Foundation, Inc. website. http://www.thalassemia.org/gohome.html.
Joint Center for Sickle Cell and Thalassemic Disorders website. http://cancer.mgh.harvard.edu/medOnc/sickle.htm.
[thal″ah-se´me-ah]
a heterogeneous group of hereditary hemolytic anemias marked by a decreased rate of synthesis of one or more hemoglobin polypeptide chains, classified according to the chain involved (α, β, δ); the two major categories are α- and β-thalassemia.
α-thalassemia (alpha-thalassemia) that caused by diminished synthesis of alpha chains of hemoglobin. The homozygous form is incompatible with life, the stillborn infant displaying severe hydrops fetalis. The heterozygous form may be asymptomatic or marked by mild anemia.
β-thalassemia (beta-thalassemia) that caused by diminished synthesis of beta chains of hemoglobin. The homozygous form is called t. major and the heterozygous form is called t. minor.
thalassemia ma´jor the homozygous form of β-thalassemia, in which hemoglobin A is completely absent; it appears in the newborn period and is marked by hemolytic, hypochromic, microcytic anemia; hepatosplenomegaly; skeletal deformation; mongoloid facies; and cardiac enlargement.
thalassemia mi´nor the heterozygous form of β-thalassemia; it is usually asymptomatic, but there may be mild anemia.
sickle cell–thalassemia a hereditary anemia involving simultaneous heterozygosity for hemoglobin S and thalassemia.
thal·as·se·mi·a
, thalassanemia (thal'ă-sē'mē-ă, thă-las-ă-nē'mē-ă),
Any of a group of inherited disorders of hemoglobin metabolism in which there is impaired synthesis of one or more of the polypeptide chains of globin; several genetic types exist, and the corresponding clinical picture may vary from barely detectable hematologic abnormality to severe and fatal anemia.
[G. thalassa, the sea, + haima, blood]
/thal·as·se·mia/ (thal″ah-se´me-ah) a heterogeneous group of hereditary hemolytic anemias marked by a decreased rate of synthesis of one or more hemoglobin polypeptide chains, classified according to the chain involved (α, β, δ); the two major categories are α- and β-thalassemia.
α-thalassemia that caused by diminished synthesis of alpha chains of hemoglobin. The homozygous form is incompatible with life, the stillborn infant displaying severe hydrops fetalis. The heterozygous form may be asymptomatic or marked by mild anemia.
β-thalassemia that caused by diminished synthesis of beta chains of hemoglobin. The homozygous form is called t. major and the heterozygous form is called t. minor.
thalassemia ma´jor the homozygous form of β, in which hemoglobin A is completely absent; it appears in the newborn period and is marked by hemolytic, hypochromic, microcytic anemia, hepatosplenomegaly, skeletal deformation, mongoloid facies, and cardiac enlargement.
thalassemia mi´nor the heterozygous form of β, usually asymptomatic, although there is sometimes mild anemia.
(thăl′ə-sē′mē-ə)
An inherited form of anemia occurring chiefly among people of Mediterranean descent, caused by faulty synthesis of part of the hemoglobin molecule. Also called Mediterranean anemia.
thal′as·se′mic adj.
[thal′əsē′mē·ə]
Etymology: Gk, thalassa, sea, a + haima, without blood
production and hemolytic anemia characterized by microcytic, hypochromic red blood cells. Thalassemia is caused by inherited deficiency of alpha- or beta-globin synthesis. See also hemochromatosis, hemosiderosis.
Beta thalassemia, clinical thalassemia, Cooley's anemia, Mediterranean anemia, thalassemia major Hematology A group of genetic diseases by underproduction of hemoglobin due to mutations in the beta globin gene, which is more common in Mediterraneans Heredity Parents are carriers–heterozygotes; one in 4 children is homozygous for the mutation and thus has full-blown disease Clinical See Anemia. Cf Sickle cell anemia.
α-thalassemia
Hemoglobin Barts Hematology An inherited condition caused by a defect in the synthesis of the Hb α chain; Hb Barts hemoglobinopathy is characterized by the presence of 4 gamma chains; it is more common in southeast Asians; the most severe form of alpha thalassemia causes stillbirth due to hydrops fetalis Heredity Parents are carriers–heterozygotes; one in 4 children is homozygous for the mutation and thus has full-blown disease Clinical Pallor, fatiguability, FTT, fever, infections, diarrhea Management Transfusions
Thalassemia major Hematology A hemoglobinopathy caused by a defect in the synthesis of Hb β chain Clinical Pallor, fatigability, FTT, fever due to infections, diarrhea, bone deformities, hepatosplenomegaly Management Transfusions, but iron overload can damage the heart, liver, and endocrine systems, ergo iron chelation–early use of deferiprone, deferoxamine ↓ transfusion-related iron overload and may protect against DM, cardiac disease, early death
δ-thalassemia
Hematology A condition characterized by a defect of Hb A2–α2δ2; because Hb A2 comprises only 3% of the circulating Hb, even its complete absence; δ-thalassemia has little clinical or hematologic impact
γ-thalassemia
Hematology A condition characterized by a defect of gamma–γ Hb chains found in Hb F–α2γ2; because Hb F is present primarily in the fetus and newborns, it is rarely seen outside of the neonatal period, but may cause transient neonatal hemolytic anemia.
, thalassanemia (thal'ă-sē'mē-ă, -ă-să-nē'mē-ă)
Any of a group of inherited disorders of hemoglobin metabolism in which there is impaired synthesis of one or more of the polypeptide chains of globin; several genetic types exist, and the corresponding clinical picture may vary from barely detectable hematologic abnormality to severe and fatal anemia. People of Mediterranean, extraction are more often affected than others by this type of anemia.
Synonym(s): thalassaemia, thalassanaemia.
Any of a group of inherited disorders of hemoglobin metabolism with impaired synthesis of one or more polypeptide chains of globin; several genetic types exist.
<a href="https://medical-dictionary.thefreedictionary.com/Thalassaemia+minor">thalassemia</a>
Barts hemoglobin
beta hemoglobinopathy
beta-delta thalassemia
beta-thalassaemia
Bite Cell
black gallstone
I know of a couple, totally unrelated and unbeknownst to them, who are silent carriers of Thalassaemia minor.
Pakistan: Genetic factor: All in the genes
But, unfortunately, when one person with thalassaemia minor carrier happens to marry another with the same diagnosis, there is a strong possibility that their child would be thalassaemia major, as happened in the case of Taneja.
' My life depends upon a monthly blood transfusion '
0] thalassaemia demonstrates variable severity, ranging from a condition similar to [beta] thalassaemia minor to something approaching thalassaemia major.
A retrospective review of homozygous haemoglobin E patients
Thal, Alan P.
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thalamencephalic
thalamencephalon
thalamic
thalamic fasciculus
thalamic nucleus
thalamic pain syndrome
thalamic peduncle
thalamic radiation
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thalamocortical
thalamocortical fibers
thalamogeniculate artery
thalamolenticular
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thalamotuberal artery
Thalassaemia minor
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thalassoplankton
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thalidomide neuropathy
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Thalassaemia Association of Malaysia
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Thalassemia Action Group
Thalassemia Clinical Research Network
thalassemia syndrome

Answer: According to the globin that is affected (alpha or beta).


Input: What does the paper aim to solve?

Context:
Paper Info

Title: Generalized Pole-Residue Method for Dynamic Analysis of Nonlinear Systems based on Volterra Series
Publish Date: March 7, 2023
Author List: Qianying Cao (from State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology), Anteng Chang (from College of Engineering, Ocean University of China), Junfeng Du (from College of Engineering, Ocean University of China), Lin Lu (from State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology)

Figure

Fig. 1: Procedure to compute the response by a combination of Volterra series and Laguerre polynomials
Fig. 2: Linear frequency response function: (a) Modulus of H 1 (ω), (b) phase angle of H 1 (ω)
Fig. 6: Comparison of h 1 (t) based on the analytical and reconstructed by Laguerre polynomials
Fig. 11: Response for Case 1: (a) comparison between the proposed method and Runge-Kutta method, (b) contribution of the three components
Fig. 18: Comparison of original excitations and reconstructed results: (a) Case 1, (b) Case 2 (c) Case 3
Fig. 19: Response to irregular excitation for Case 1: (a) comparison between the proposed method and Runge-Kutta method, (b) contribution of the three components
Fig. 23: Input-output dataset used to identify Volterra series: (a) input excitation, (b) output response
Fig. 26: Comparison of responses between the predicted and numerical results: (a) response to regular excitation, (b) response to irregular excitation
Parameter values of the irregular excitation

abstract

Dynamic systems characterized by second-order nonlinear ordinary differential equations appear in many fields of physics and engineering. To solve these kinds of problems, time-consuming stepby-step numerical integration methods and convolution methods based on Volterra series in the time domain have been widely used.
In contrast, this work develops an efficient generalized pole-residue method based on the Volterra series performed in the Laplace domain. The proposed method involves two steps: (1) the Volterra kernels are decoupled in terms of Laguerre polynomials, and (2) the partial response related to a single Laguerre polynomial is obtained analytically in terms of the pole-residue method.
Compared to the traditional pole-residue method for a linear system, one of the novelties of the pole-residue method in this paper is how to deal with the higher-order poles and their corresponding coefficients. Because the proposed method derives an explicit, continuous response function of time, it is much more efficient than traditional numerical methods.
Unlike the traditional Laplace domain method, the proposed method is applicable to arbitrary irregular excitations. Because the natural response, forced response and cross response are naturally obtained in the solution procedure, meaningful mathematical and physical insights are gained. In numerical studies, systems with a known equation of motion and an unknown equation of motion are investigated.
For each system, regular excitations and complex irregular excitations with different parameters are studied. Numerical studies validate the good accuracy and high efficiency of the proposed method by comparing it with the fourth-order Runge-Kutta method.

Introduction

Most real dynamic systems, as encountered in mechanical and civil engineering, are inherently nonlinear and include geometric nonlinearities, nonlinear constitutive relations in material or nonlinear resistances, etc. . Nonlinear problems are attracting increasing attention from engineers and scientists.
This work focuses on solving nonlinear system vibration problems, i.e., computing transient responses of nonlinear oscillators under arbitrary irregular excitations based on a combination of a pole-residue operation and Volterra series. Because Volterra series are single-valued, the scope of the present study is restricted to nonlinear behaviours without bifurcations .
To analyse nonlinear vibration problems, researchers have performed extensive studies and developed various mathematical methods. Popular methods include step-by-step numerical integration methods in the time domain, such as the Runge-Kutta method. This kind of method not only requires a small time-step resolution for obtaining high-precision solutions but also is prone to numerical instability .
For a long response with small time steps, the time domain methods are very costly in computational time. Volterra series is another widely used method, which is the extension of the Duhamel integral for linear systems . Volterra series can reproduce many nonlinear phenomena, but they are very complex due to higher-dimensional convolution integrals .
Since 1980's, significant progress has been made in the general area of the Volterra series. The reader is referred to Ref. for a quite thorough literature review on the relevant topics. After 2017, most papers focus on Volterra series identification. De Paula and Marques proposed a method for the identification of Volterra kernels, which was based on time-delay neural networks.
Son and Kim presented a method for a direct estimation of the Volterra kernel coefficients. Dalla Libera et al. introduced two new kernels for Volterra series identification. Peng et al. used the measured response to identify the kernel function and performed the nonlinear structural damage detection. Only a few papers concentrated on simplifying the computation of convolution integrals.
Traditional methods for computing convolution integrals involved in the Volterra series have been performed in three distinct domains: time, frequency and Laplace. The time domain method based on Volterra series refers to discrete time convolution methods, which also suffer computational cost problems .
Both the frequency domain method and the Laplace domain method based on the Volterra series consist of three steps: (1) Volterra series are transformed into an algebraic equation in the frequency domain or Laplace domain; the algebraic equation is solved by purely algebraic manipulations; and (3) the solution in Step ( ) is transformed back to the time domain.
Many researchers have used the frequency domain method to compute the responses of nonlinear systems. Billings et al. developed a new method for identifying the generalized frequency response function (GFRF) of nonlinear systems and then predicted the nonlinear response based on these GFRFs. Carassale et al. introduced a frequency domain approach for nonlinear bridge aerodynamics and aeroelasticity.
Ho et al. computed an output frequency domain function of a nonlinear damped duffing system modelled by a Volterra series under a sinusoidal input. Kim et al. identified the higher order frequency response functions by using the nonlinear autoregressive with exogenous input technique and the harmonic probing method.
This type of frequency domain method is much more efficient than the time domain method due to the fast Fourier transform algorithm. However, the frequency domain method not only is limited by frequency resolutions but also suffers from leakage problems due to the use of discrete Fourier transforms. In addition, the frequency domain method calculates only a steady-state response.
A natural response generated by initial conditions and a cross response caused by interactions between a system and an excitation are ignored. In contrast, the Laplace domain method can calculate all response components because initial conditions are considered in the computational procedure. However, it has been restricted to analytical operations for simple excitations, such as sinusoidal excitations and exponential excitations .
The proposed method falls into the category of the Volterra series method computed in the Laplace domain. Unlike the traditional Laplace domain method, the proposed method is applicable to arbitrary irregular excitations. Because the proposed method follows a similar path as a pole-residue method for linear systems , the proposed method to solve nonlinear system vibration problems is called the generalized pole-residue method.
The main concept of the pole-residue method developed by Hu et al. was that the poles and residues of the response could be easily obtained from those of the input and system transfer function to obtain the closed-form response solution of linear systems. This method included three steps: (1) writing the system transfer function into pole-residue form; (2) writing the excitation into pole-residue form by the Prony-SS method; (3) computing the poles and residues of the response by an algebraic operation based on those from system and excitation.
Compared to Hu et al. , which was regarded as an efficient tool to compute responses of linear systems, the generalized pole-residue method in this paper is introduced to compute responses of nonlinear systems. The proposed method involves two steps: (1) the Volterra kernels are decoupled in terms of Laguerre polynomials, and (2) the partial response related to a single Laguerre polynomial is obtained analytically in terms of the pole-residue method.
Compared to the traditional pole-residue method for a linear system, one of the novelties of the generalized pole-residue method is how to deal with the higher-order poles and their corresponding coefficients. Similar to the Taylor series, the Volterra series representation is an infinite series, and convergence conditions are needed to assure that the representation is meaningful.
Because the proposed method is based on the Volterra series, only the system with convergent Volterra series representation can be treated by the proposed method. The paper is organized as follows. In Section 2, the nonlinear response is modelled by a Volterra series, and Volterra kernel functions are decoupled by Laguerre polynomials.
Then, the pole-residue method for computing explicit responses is developed in Section 3. Numerical studies and discussions are given in Section 4. Finally, the conclusions are drawn in Section 5.

Response calculation based on Volterra series

A nonlinear oscillator, whose governing equation of motion is given by where z(t, y, ẏ) represents an arbitrary nonlinear term; m, c, and k are the mass, damping and linear stiffness, respectively; y(t), ẏ(t) and ÿ(t) are the displacement, velocity and acceleration, respectively; and f (t) is the time-dependent excitation.
If the energy of excitation f (t) is limited, the nonlinear response under zero initial conditions (i.e., zero displacement and zero velocity) can be represented by the Volterra series : where N is the order of Volterra series and In Eq. 3, h 1 (τ ) is called the first-order Volterra kernel function, which represents the linear behaviour of the system; h n (τ 1 , . . .
, τ n ) for n > 1 are the higher-order Volterra kernel functions, which describe the nonlinear behaviour of the system. The complete formulation of y(t) includes infinite series where the labour of calculating the n th term increases quickly with the growth of n. Fortunately, the response accuracy may be ensured by the first several order Volterra series.
This is proved here in numerical studies. The commonly known Laguerre polynomials are represented as : where p i is the order of the Laguerre polynomials and a i is the damping rate. The Laguerre polynomials satisfy the orthogonal relationship expressed as: By using Laguerre polynomials, the Volterra kernel function h n (t 1 , . . .
, t n ) in Eq. 3 can be decoupled as follows : where the coefficient is computed resorting to the orthogonal relationship in Eq. 5: Substituting Eq. 6 into Eq. 3 yields . . . The above operation that uses the Laguerre polynomials to decouple Volterra higher order kernel functions has been well-developed.
The reader is referred to Refs. for details about the adopted technique. After decoupling Volterra higher order kernel functions in time, one can regroup Eq. 8 into: . . . By denoting Eq. 9 becomes The above procedure to compute the nonlinear response by a combination of Volterra series and Laguerre polynomials is schematically shown in Fig. .
Volterra kernel functions h n (t 1 , . . . , t n ) can be obtained by either an equation of motion or measured input-output signals. To derive a closedform solution of the response, we must obtain a closed-form solution of x i (t) first. In the following presentation, a closed-form solution of the aforementioned x i (t) and y n (t) is derived by using the pole-residue method.
3. Pole-residue method for calculating x i (t) and y n (t) Performing the Laplace transform of x i (t) in Eq. 10 yields where in which Eq. 13 includes a single pole and several higher-order poles. For k = 0, −a i is a single pole, and b p i (0) is a corresponding coefficient, namely, the residue. For k > 0, −a i are higher-order poles, and b p i (k) are corresponding coefficients.
For an irregular excitation signal f (t) of a finite duration of T , it can always be approximated into a pole-residue form by using the complex exponential signal decomposition method-Prony-SS : where N ℓ is the number of components; α ℓ and λ ℓ are constant coefficients, which either are real numbers or occur in complex conjugate pairs.
We define λ ℓ = −δ ℓ + iΩ ℓ , where Ω ℓ is the excitation frequency and δ ℓ is the damping factor of the ℓ th component. We denote α ℓ = A ℓ e iθ ℓ , where A ℓ is the amplitude and θ ℓ is the sinusoidal initial phase in radians. Taking the Laplace transform of Eq. 15 yields Note that the concept of the Prony-SS method is similar to that of a principal component method.
A smooth excitation usually requires just several terms to achieve a good approximation. For high irregular loadings, including more terms would achieve a better approximation. Substituting Eqs. 13 and 16 into Eq. 12 yields Expressing xi (s) in its pole-residue form yields where λ ℓ are simple poles, and the corresponding residues are easily obtained by
and −a i are higher-order poles, and the corresponding coefficients are firstly derived as: By taking the inverse Laplace transform of Eq. 18, a closed-form solution is obtained: Substituting Eqs. 11 and 21 into Eq. 2 yields Theoretically speaking, the proposed method for deriving the closed-form solution of the nonlinear response is applicable to any order of the Volterra series.
For practical engineering, usually only the first several order responses dominate. By setting up N = 2, Eq. 22 can be simplified into three components: where the natural response, which is only related to system poles, is given by and the cross response, which is related to both system poles and excitation poles, is given by
and the forced response, which is related only to excitation poles, is given by The first term in Eq. 26 is the first-order forced response governed by the excitation frequency, i.e., the imaginary part of the pole λ ℓ . The second term corresponds to the second-order nonlinear forced response, which includes the sum frequency and difference frequency responses governed by λ ℓ + λ j .
Eq. 26 straightforwardly offers visible information about the possible nonlinear vibrations by the cooperation of excitation frequencies. Particularly, consider a sinusoidal excitation f (t) = sin ω r t, which can be expressed as f (t) = γe λt + γ * e λ * t , where γ = −0.5i and λ = iω r . Substituting these values into Eq.
26, the second term of Eq. 26 is simplified as where the first term is the difference frequency response, and the second term is the sum frequency response.

Numerical studies

In practical engineering, some systems have an accurate equation of motion. Additionally, some systems have difficulty constructing their equations of motion because of complex nonlinear dynamic behaviours and uncertain system parameters. In this article, a system with a known equation of motion is called a known system, and a system with an unknown equation of motion is called an unknown system for simplicity.
In this section, two numerical studies are presented. The first study verifies the proposed method using a known nonlinear oscillator, and the second study demonstrates the applicability of the proposed method to an unknown system. Throughout the numerical studies, the unit system is the metre-kilogramme-second (MKS) system; for conciseness, explicit units for quantities are omitted.

A known nonlinear system

This study chooses a nonlinear oscillator written as: where mass m = 1, damping c = 1, linear stiffness k 1 = 10, quadratic stiffness k 2 = 20 and cubic stiffness k 3 = 20. It is a case that has been studied in a previously published article . The linear natural frequency of the system ω 0 = k 1 /m = 3.16 and the damping ratio ζ = c/(2mω 0 ) = 15.8%.
This kind of oscillator occurs in many engineering problems, such as a model of fluid resonance in a narrow gap between large vessels . In the model, k 1 y represents the linear restoring force of the fluid, and k 2 y 2 and k 3 y 3 are respectively the quadratic and cubic nonlinear restoring forces of the fluid.

Volterra kernel functions

Generally, the first several order responses dominate the total response of a system. Hence, the order of the Volterra series in Eq. 22 is chosen to be 3, namely, N = 3. For computing the first three order responses from Eq. 22, the first three order Volterra kernel functions need to be known. Since Volterra kernel functions and corresponding frequency response functions are related by a specific Fourier transform pair, we can first write the first three orders of frequency response functions directly from Eq. 28.
Then, Volterra kernel functions are obtained by the inverse Fourier transform. Based on the harmonic probing algorithm , the linear frequency response function (LFRF) H 1 (ω), the quadratic frequency response function (QFRF) H 2 (ω 1 , ω 2 ) and the cubic frequency response function (CFRF) H 3 (ω 1 , ω 2 , ω 3 ) are analytically given by:
and Figures show H 1 (ω), H 2 (ω 1 , ω 2 ) and H 3 (ω 1 , ω 2 , ω 3 ), respectively, which agree well with those reported in Ref. . As expected, the modulus of H 1 (ω) in Fig. peaks near the linear natural frequency ω 0 , and the phase angle decreases monotonically from 0 to -π with increasing frequency.
Figure shows the sum frequency QFRF, where the energy converges along the line of ω 1 +ω 2 ≈ ω 0 . Therefore, when the sum frequency of a two-tone excitation equals the linear resonant frequency, the second-order response may reach its maximum. Additionally, those pairs of excitations in line ω 1 + ω 2 ≈ ω 0 may produce non-negligible vibration magnitudes due to second-order nonlinear effects.
For the difference frequency QFRF in Fig. (b), the energy converges along two main lines, i.e., ω 1 ≈ ω 0 and ω 2 ≈ ω 0 . Figures show moduli of H 3 (ω, ω, ω) and H 3 (ω, ω, −ω), which are diagonal terms of the sum frequency CFRF and the difference frequency CFRF, respectively. While the modulus of H 3 (ω, ω, ω) peaks near ω ≈ ω 0 /3 and ω 0 , that of H 3 (ω, ω, −ω) peaks near ω ≈ ω 0 with a small hump around ω ≈ ω 0 /2.
Values at ω ≈ ω 0 /3 and ω 0 /2 may be magnified by higher-order stiffness terms in Eq. 28. By performing the inverse fast Fourier transform to Eqs. 29-31, the corresponding linear impulse response function h 1 (t), quadratic impulse response function h 2 (t 1 , t 2 ) and cubic impulse response function h 3 (t 1 , t 2 , t 3 ) are obtained.
Here, h 1 (t) and h 2 (t 1 , t 2 ) are plotted in Figs. , respectively, and h 3 (t, t, t) is shown in Fig. . In the numerical implementation, Eqs. 29-31 have been utilized with the frequency interval ∆ω = 0.1, number of frequency components N n = 1025, and cut-off frequencies 102.4 and −102.4. For decoupling Volterra kernel functions by using Laguerre polynomials, the damping rate and number of Laguerre polynomials for each order Volterra kernel function need to be determined (see Eqs. 4 and 6).
In this example, we set a 1 = a 2 = a 3 = 2 and R 1 = R 2 = R 3 = 24 because coefficients c p 1 ...pn become very small when R n > 24, n = 1, 2, 3. According to Eq. 7, the coefficients of the first three order Volterra kernel functions are calculated, which are shown in Figs. 9 and 10. For convenience, Fig. plots only c p 1 p 2 p 3 for p 3 = 0.
With the increase of the order of Laguerre polynomials, coefficients in Figs. 9 and 10 gradually decrease, which illustrates how the first several orders of Laguerre polynomials dominate all orders of the Volterra kernel function. With the known Laguerre polynomials and corresponding coefficients, Volterra kernel functions are reconstructed by Eq. 6.
For comparison, reconstructed Volterra kernel functions are also plotted in Figs. . The reconstructed results agree well with the analytical values, which verifies the accuracy of the decomposition.

Sinusoidal excitation

From Eq. 28, we consider a sinusoidal excitation where A and Ω are the amplitude and the frequency, respectively. Five cases of A and Ω are shown in Table . Excitation frequencies in Cases 1 and 2 are larger than the linear natural frequency (ω 0 ≈ 3.16), those in Case 3 are very close to ω 0 , and those in Cases 4 and 5 are smaller than ω 0 .
All cases have same amplitudes. The poles of a sinusoidal excitation are λ 1,2 = ±iΩ, and the residues are α 1,2 = ∓iA/2. Numerical values of excitation poles and residues for different cases are listed in Table . Table : Parameter values, poles and residues of the sinusoidal excitation Substituting poles and residues of the excitation, as well as those of the system into Eqs.
20 and 19, response coefficients β p i ,k corresponding to system poles −a i and response coefficients γ p i ,ℓ corresponding to excitation poles λ ℓ are calculated, respectively. According to Eq. 22, the first three orders of responses for each case in Table are calculated. Figures )-15(a) show the comparison of responses obtained by the proposed method and the fourth-order Runge-Kutta method with ∆t = 10 −4 .
For Cases 1 and 2, the first-order responses agree well with the total responses obtained by the Runge-Kutta method, and the higher-order responses only slightly improve the transient parts. For Cases 3-5, the sum of the first three orders of responses is in good agreement with the Runge-Kutta solution.
When the response nonlinearity increases, higher-order responses need to be considered. In other words, the proposed method can accurately compute the nonlinear responses by choosing a small number N of Volterra series terms. Figures )-15(b) show the contributions of the three response components for the five cases.
In each case, the first-order response is the most dominant component, and the contributions of secondand third-order responses are much less than those of the first-order response. Especially for Cases 1 and 2, whose excitation frequencies are far from the linear natural frequency, second-and thirdorder responses are close to zero.
This may be because the QFRF and CFRF approach zero when the frequency is larger than 4 rad/s (see Figs. ). Furthermore, the mean values of the first-order responses are approximately zero, and those of the second-order responses are always smaller than zero, which are the difference frequency components in Eq. 27.
Moreover, it is clearly observed that second-order responses for Cases 3-5 exhibit a periodic oscillation with a period near half of that for the first-order response, which is excited by the sum frequency component of the excitation (see second part of Eq. 27). Compared with steady-state solutions of first-and second-order responses, those of third-order responses in Cases 3-5 are no longer single regular motions.
By performing the FFT, frequency spectra of these three third-order responses are shown in Fig. . We find that these three third-order responses are all dominated by their own fundamental harmonic component and the third harmonic (triple frequency) component. Figure shows the computational time to calculate the response of the oscillator for Case 1 by the proposed method, the fourth-order Runge-Kutta method and the convolution method.
The proposed method, which has an explicit solution, is much more efficient in computational time than the latter two methods, which need small time steps to obtain high-precision solutions. In particular, the efficiency of the proposed method increases with the length of the response time. Computation time (sec.)
t=0.02s Convolution, t=0.02s Convolution, t=0.001s Runge-Kutta, t=0.001s Fig. : Comparison of computation efficiency of the proposed method, the fourth-fifth order Runge-Kutta method and the convolution method regular loading in Case 1

Irregular excitation

In Eq. 28, considering an irregular excitation consisting of several cosine functions where N f is the number of cosine components; A n , Ω n and θ n are the amplitude, frequency and phase angle of the n th component, respectively. Table lists three cases of these parameters. In each case, the amplitudes of all components are the same, and phase angles θ n uniformly distributed between 0 and 2π are randomly generated.
To decompose the excitation into a pole-residue form, the Prony-SS method is used, whose concept is similar to that of a principal component method. The readers are referred to Ref. for details. The chosen rank of each case is also shown in Table . Figure shows the comparison of original excitations and reconstructed results of these three cases, which all have excellent agreement.
show the results computed by the fourth-order Runge-Kutta method. In all cases, the sums of the first three orders of responses agree well with those obtained by the Runge-Kutta method. The contributions of the first three orders of responses for each case are plotted in Figs. )-21(b). Similarly, the system vibration is dominated by the first-order response.
However, the contributions of second-and third-order significantly grow with increasing excitation magnitude and frequency number. Furthermore, when the magnitude of the nonlinear response becomes large, sharp troughs are present. This phenomenon may be induced by the nonlinear stiffness. While the first-order response fails to capture these troughs, the higher-order responses successfully capture these troughs.
Figure plots the computational time to calculate the response of the oscillator for the irregular loading in Case 1 by the proposed method and the fourth-fifth order Runge-Kutta method, respectively. While the fourth-fifth order Runge-Kutta method is more efficient under a small response length, the proposed method becomes much more efficient when the response length is larger than about 130 s.
In addition, the proposed method obtains the explicit response solution, so one can directly obtain the response value at a specific time t p instead of integrating from 0 to t p for traditional numerical methods. Computation time (sec.) Proposed, t=0.02s Runge-Kutta, t=0.001s Fig. : Comparison of computation efficiency of the method and the fourth-fifth order Runge-Kutta method for irregular loading in Case 1

An unknown nonlinear

To check the applicability of the proposed method to an unknown nonlinear system, a known input excitation and its corresponding response are used to identify its Volterra kernel functions. When the Volterra kernel functions are known, we can follow the procedure in Section 4.1 to predict system responses.
In this study, the input excitation is white noise with a constant power spectrum S 0 = 0.001, and the corresponding response is obtained by solving Eq. 28 by the fourth-order Runge-Kutta method, which is shown in Fig. . From Section 4.1, we determine that the sum of the first two orders of responses agrees well with the total response.
In this study, the order of Volterra series N is chosen to be 2, damping rates of Laguerre polynomials are a 1 = a 2 = 2, and numbers of Laguerre polynomials are R 1 = R 2 = 24. To estimate the first two orders of Volterra kernel functions, a matrix equation is constructed using excitation data and response data.
By using the least square method to solve this matrix equation, coefficients c p 1 and c p 1 p 2 in Eq. 8 are identified. Figure plots c p 1 and c p 1 p 2 , respectively, which have good agreement with the exact results in Fig. . Then, the first two order Volterra kernel functions are constructed by Eq. 6.
Compared with the exact results in Figs. , the identified Volterra kernel functions in Fig. completely agree well with the exact solutions. Note that the white noise excitation, which can excite more frequency components of the response, is chosen to obtain good Volterra kernel functions. A regular excitation f (t) = sin(πt) and an irregular excitation f (t) = N f n=1 A n cos(Ω n t + θ n ) with A n = 0.3 and Ω n varying from 0 to 40 with equal interval 1 are chosen as input excitations.
The predicted responses, along with results obtained by the fourth-order Runge-Kutta method, are shown in Fig. . In both cases, the proposed method accurately predicts system responses. As presented in Eq. 23, a nonlinear response is the sum of three terms: natural response y s (t), forced response y f (t) and cross response y c (t).
These individual terms, as well as their sum to two excitations, are shown in Figs. 27 and 28, respectively. As shown in Figs. and 28, both first-and second-order responses include the natural response y s (t) and the forced response y f (t), but the cross response y c (t) only exists in second-order responses.
When t becomes larger, both y s (t) and y c (t) diminish due to the presence of system damping, and the total response is entirely governed by y f (t). Moreover, we notice some features at t = 0 for these components, including y s (0) = −y f (0) for the first-order response and y s (0) + y f (0) = −y c (0) for the second-order response, which are due to imposed zero initial conditions.

Conclusions

Considering arbitrary irregular excitations, an efficient generalized pole-residue method to compute the nonlinear dynamic response modelled by the Volterra series was developed. A core of the proposed method was obtaining poles and corresponding coefficients of Volterra kernel functions, then those of each order response modelled by each order Volterra series.
Once the poles and corresponding coefficients of Volterra kernel functions and excitations were both available, the remaining derivation could follow a similar pole-residue method that had been developed for ordinary linear oscillators. To obtain the poles and corresponding coefficients of Volterra kernel functions, two steps were included: (1) using Laguerre polynomials to decouple higher-order Volterra kernel functions with respect to time and (2) obtaining poles and corresponding coefficients of Laguerre polynomials in the Laplace domain.
Because the proposed method gave an explicit, continuous response function of time, it was much more efficient than traditional numerical methods. Moreover, many meaningful physical and mathematical insights were gained because not only each order response but also the natural response, the forced response and the cross response of each order were obtained in the solution procedure.
To demonstrate that the proposed method was not only suitable for a system with a known equation of motion but also applicable to a system with an unknown equation of motion, two numerical studies were conducted. For each study, regular excitations and complex irregular excitations with different parameters were investigated.
The efficiency of the proposed method was verified by the fourth-order Runge-Kutta method. This paper only computes the response under zero initial conditions. The response under non-zero initial conditions will be investigated in our future work.

Answer: The paper aims to solve nonlinear system vibration problems efficiently.