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14.2: Bone

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    We tend to think of our bones as dry—a Halloween skeleton—chalky, dead tissue. But bone is well supplied with blood, and minerals constantly move in and out of bone. Some bones contain red marrow which makes red blood cells, some white blood cells, and platelets. The skeleton is very much alive.

    Besides its use as support, protection, movement, and blood-cell production, bone is a mineral reserve. About 99% of the body’s calcium, 85% of the body’s phosphorus, and 60% of the body’s magnesium are in bone. These minerals are crucial to body chemistry. Without calcium, blood can’t clot, muscle can’t contract, and a nerve cell can’t send its message. Phosphorus is a key part of ATP (adenosine triphosphate), and magnesium is needed to regenerate ATP.

    Very small amounts of calcium, phosphorus, and magnesium are needed for these uses, but their presence in bone is crucial, and the body is assured of never running out. Bone minerals are readily mobilized to make up for any shortfall.

    Bone is about 30% protein (collagen) and about 70% minerals. It’s sort of like reinforced concrete —protein for tensile strength, and minerals for compressive strength. Taking out the minerals is a popular experiment in a child’s science class: Put a chicken bone in vinegar; the acid (acetic acid in vinegar) dissolves the minerals; after a few weeks, only the protein is left—demonstrated by bending the bone in half.*

    When a bone is said to be dense, it’s dense with minerals. Bones aren’t really solid. The outside is smooth and compact, but the inside is porous, much like styrofoam. It’s very well engineered. The outer hardness of a leg bone provides the strength of a tube (a tube is almost as strong as a solid rod of the same diameter). The porous inside adds strength without adding much weight. The protein gives it some flexibility.

    Stimulated by growth hormone, bone grows fast during childhood and adolescence. Bone density increases until about age 30, but most of that density is acquired before age 20. Generally speaking, bone density stays about the same from age 30 to 50 and then falls. A key factor in the development of osteoporosis is a person’s bone density in early adulthood when bone mass reaches its peak (peak bone density).

    In dwarfism, the pituitary gland doesn’t make enough growth hormone during childhood, and the person is abnormally short. Growth hormone (a protein) used to be extracted from human pituitary glands, and the limited supply was used to treat children with dwarfism. Some of these children came down, many years later, with a fatal brain disease caused by an infectious agent in the extracted pituitary glands. Production of human growth hormone by biotech now eliminates this risk and makes more hormone available.

    Some parents request this hormone for their children to make them taller, even though the children are of normal height and have normal amounts of growth hormone. Treatment takes about 6 years, and is very expensive. Growth hormone is legitimately available by prescription, but most physicians don’t approve of its use for this purpose.

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    *When soup bones are boiled, collagen (a protein) dissolves, forming gelatin. This is why soup stock gels when refrigerated. Gelatin (for Jell-O, etc.) is typically made by extracting it from pig skin, animal bones, etc.

    Bone Nutrients

    Vitamin D

    Many nutrients are important in bone development and maintenance, but the focus is on vitamin D and calcium, since these play the biggest roles. Vitamin D is called the sunshine vitamin—it can be made in skin by the action of ultraviolet (UV) light from the sun (or a sunlamp). UV light converts a vitamin D precursor (made in the body from cholesterol) into vitamin D (see Figure 14.2). This occurs in other animals as well.

    Chickens running around the farmyard make enough vitamin D from sun exposure. After inexpensive sources of vitamin D were found, it could be added to the diet of domesticated animals (including pets), and they could be raised entirely indoors. Foods like eggs and poultry then became more plentiful and inexpensive, improving the diet of many people.

    One might argue that vitamin D isn’t really a vitamin, since we can make what we need if we get enough sun. But without enough sun, vitamin D is needed in the diet.

    It’s hard to say how much sun is enough. The amount of UV radiation that reaches the skin varies with time of day, latitude, cloud cover, air pollution, clothing style and custom, and amount of sunscreen put on the skin. Also, those with lighter skin need less exposure. Production of vitamin D in the skin is less efficient in older adults.

    Another complicating factor is that we are advised to wear sunscreen to protect against skin cancer, and sunscreen lessens vitamin D production. Due to the uncertainty of the amount of vitamin D production from sun exposure, there are Recommended Dietary Allowances (RDAs) for vitamin D.

    Once vitamin D is made in the skin or taken in from the diet, its chemical structure is altered in the kidney and liver to an activated form of vitamin D (see Figure 14.2). Thus, people with liver or kidney disease may have symptoms of a deficiency, even if they get enough vitamin D. They could then be prescribed the activated version of vitamin D.

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    Figure 14.2: Vitamin D Activation

    Vitamin D is needed for bone to grow correctly. It’s needed to absorb calcium from the intestine and to mineralize bone (vitamin D turns on the gene for a calcium-binding protein used to take in calcium). Without enough vitamin D, bones aren’t fully mineralized, making them weak and easily bent. (In addition to bone health, vitamin D plays a role in other functions, e.g., immunity.)

    Severe vitamin D deficiency causes osteomalacia when it occurs in adults, and rickets when it occurs in children. A child with rickets has bowed legs from the weight of the body pressing down on the developing leg bones.

    The bone deformities can cause serious problems. Bent rib bones can cause a permanent “sunken chest” in children, making them more prone to lung disease. A permanently narrowed pelvic bone in girls can be a problem in childbirth. Osteomalacia (severe vitamin D deficiency in adults) can cause painful fractures, especially in the spine and pelvis.

    Rickets and osteomalacia were common in industrialized cities during the 17th century. Buildings that lined the narrow streets and the heavy soot in the air blocked sunlight. The diet was also poor in vitamin D.

    Rickets was so common that it was thought to be normal. Before vitamin D (and its relationship to rickets) was discovered, it was found that cod-liver oil (rich in vitamin D) could prevent rickets. In the 19th century, many a mother forced her children to swallow a squirt of smelly cod-liver oil every day.

    In tropical countries, vitamin D deficiency rarely occurs, regardless of diet, unless people completely avoid the sun or use sunscreen. In the practice of purdah by some cultures, women cover themselves head to toe whenever they go outdoors. This puts them at risk for vitamin D deficiency unless they get it in their diet.

    Vitamin D is found naturally in such foods as egg yolk, butter, and fish, and is added to foods such as margarine, milk, and some ready-to-eat breakfast cereals. Fatty fish such as salmon, tuna, and mackerel are particularly rich sources. Also, many people take vitamin supplements and/or eat fortified foods that include vitamin D.

    It’s advised that infants who are exclusively breast-fed be given a vitamin D supplement. (“Natural” milk— human and cow’s milk alike— has very little vitamin D. Infant formula and cow’s milk are fortified with vitamin D.)

    People taking vitamin D supplements should be warned against taking doses higher than the RDA. Excessive amounts can be very toxic—especially in young children—and can cause damaging calcium deposits in soft tissues and damage to the kidneys and cardiovascular system. Vitamin D isn’t made in toxic amounts by sun exposure).

    Vitamin D intake from the diet and/or supplement is especially important for African-Americans. Many of them are lactose intolerant, and may avoid milk for this reason. However, as mentioned in Chapter 6, many people with lactose intolerance can avoid symptoms by consuming milk or milk products in small doses and with meals. Lactose-free milk is also available.

    It takes more sun exposure to form vitamin D in dark skin because it has a lot of melanin (a brown/ black pigment)—a natural sunblock against the sun’s UV light (Chap. 12). This situation is made worse by limited sun exposure due to air pollution and by less skin exposure during the cold seasons.

    For much of human history, Africans had strong and prolonged sun exposure because they lived at latitudes near the equator. They made ample vitamin D in their skin, while their large amounts of melanin protected them from the sun’s damaging effects (e.g., skin cancer).

    In contrast, Eskimos had little exposure to the sun, because of heavy clothing and living at an extreme northern latitudes. However, they weren’t dependent on sun exposure for vitamin D. Their diet was rich in oils from seafood (e.g., fatty fish, seals)—oils that are rich in vitamin D.

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    Calcium

    The Halloween skeleton is chalky indeed. Both chalk (limestone) and the skeleton are rich in calcium. Calcium exists in nature only in combination with other elements. In chalk, it’s calcium carbonate (CaCO3). In bone, it’s mainly hydroxyapatite [Ca10(PO4)6(OH)2].

    Bone is a calcium reserve. Calcium is kept at a constant level in the blood. When it goes up, one hormone (calcitonin) takes calcium out of blood and deposits it in bone. When it falls, another hormone (parathormone) draws calcium out of bone and into the blood (see Figure 14.3).

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    Figure 14.3:Maintaining Blood-Calcium

    Problems occur when blood-calcium goes outside of its narrow range. For example, nerve cells become more sensitive with an abnormal drop in blood-calcium. This can cause a rapid stimulation of muscle nerves, such that the muscle cramps or “freezes” in contraction.

    Some popular articles falsely attribute this to a deficiency of calcium in the diet. In fact, dietary calcium doesn’t ordinarily alter blood-calcium. About 99% of the body’s calcium is in bone, and this calcium is easily mobilized to make up for any shortfall in the blood. Abnormal blood-calcium is usually from a problem with either vitamin D or one of the hormones that control the movement of calcium in and out of bone.

    Bone is constantly altered throughout life, with calcium continually moving in and out of bone to be deposited where it’s needed. Small differences in the rate in which calcium moves in or out can have big effects on bone density. Stress on a bone stimulates calcium to move in, making the bone denser. Less stress on a bone (as when people become bed-ridden or when astronauts escape the pull of gravity) causes calcium loss.

    During childhood, bone grows slower if the diet is deficient in calcium. If this persists through puberty, children stay shorter. This could be an adaptation to an inadequate diet, since less calcium is needed to maintain a shorter skeleton.

    The adaptation is temporary—if the next generation has a good diet, the children will grow to their full genetic potential. In Japan, nutrition improved right after World War II, and the children then grew several inches taller than their parents.

    Severe calcium deficiency causes rickets in children. Although vitamin D is needed to properly mineralize bone, it can’t do so if there isn’t enough calcium. The outcome (rickets) is the same, whether it’s vitamin D or calcium that’s deficient. (Vitamin D deficiency is the most common cause of rickets.)

    The main source of calcium in the U.S. diet is milk and milk products. It’s an especially good source because the vitamin D, protein, and lactose in milk promote calcium absorption. In the U.S., calcium intake generally is related to the amount of milk or milk products in the diet.

    Some ethnic foods have a fair amount of calcium. Traditional Mexican corn tortillas are rich in calcium. Corn itself isn’t rich in calcium, but it’s soaked in limewater (a solution of calcium oxide) before being ground to make the tortillas. This calcium-fortified cornmeal is called masa harina.

    Traditional soybean curd (tofu) is made rich in calcium by processing in a calcium-rich solution (some brands of “organic tofu” aren’t made this way). Sesame seeds are rich in calcium, and a paste of ground sesame seeds is the main ingredient in halvah, a Turkish confection. Sesame paste is also added to Middle-Eastern dishes such as hummus. Bones (e.g., in canned sardines and salmon, or calcium leached from bones boiled in an acidic liquid) also provide calcium.

    Where do adult animals get their calcium? Calcium-rich antlers are shed onto the forest floor where small animals eat them. When animals need a lot of calcium but eat only low-calcium foods, they eat huge amounts, e.g., cows spend virtually all their waking hours munching on grass or hay.

    The amount of calcium that’s absorbed in the intestine varies. Spinach has a fair amount of calcium, but also has lots of oxalic acid, which binds to calcium and prevents its absorption. In contrast, vitamin D promotes calcium absorption.

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    Absorption goes up as a long-term adaptation to a low-calcium diet and during early childhood, pregnancy, and breast-feeding. Absorption goes down in older adults. This variability in absorption is taken into account in setting the recommended amounts of dietary calcium.

    It’s important to get enough calcium, especially during childhood, adolescence, and early adulthood, when bone is growing and becoming more dense. Although the calcium recommendations are the same for both genders, most people over age 11 don’t get enough calcium. Females are particularly low; they drink less milk and eat less food than males the same age. The more we eat, the more nutrients we take in.

    Try to get your calcium from a good diet. Calcium supplements can cause constipation and interfere with the absorption of other essential minerals (e.g., iron). Supplements vary in the amount of calcium (Chap. 3). To be effective, the supplement must dissolve in the stomach (Chap. 6 tells how to test for this). Taking ground-up bone (bone meal) isn’t advised, since it can be contaminated with lead or other toxic elements. Bone meal may be more “natural,” but it also can be naturally contaminated.

    Bone takes in lead (withdraws it from blood), which lessens lead’s toxic effects. Bone also takes in strontium-90, a radioactive element present in fallout from nuclear explosions. Strontium is chemically similar to calcium (both atoms have 2 electrons in their outer shell; Chap. 3), so the body handles it like calcium and stores it in bone.

    Since blood cells are made in bone marrow, radioactive strontium-90 in the bone raises the risk of leukemia (cancer of white blood cells). In the case of radioactive fallout, ingesting a lot of calcium reduces the amount of strontium-90 absorbed from contaminated food—calcium and strontium compete for attachment to the calcium-binding protein that brings them in from the intestine.

    Vitamin C

    Vitamin C (ascorbic acid) is needed to make the protein collagen, a component of bone and connective tissue (e.g., in skin and gums).* Delayed wound healing (collagen is a part of scar tissue), bleeding gums, and loose teeth are classic signs of scurvy (severe C deficiency). Keep in mind that these symptoms don’t necessarily indicate a vitamin C deficiency, e.g., the most common cause of bleeding gums and loose teeth is periodontal disease caused by poor dental hygiene.

    Foods like citrus fruits, strawberries, broccoli, red and green peppers, and tomatoes are rich sources of vitamin C. Potatoes have only modest amounts, but we eat so much of them (e.g., french fries) that they are a major source. Vitamin C is also added to many foods, especially fruit-flavored drinks.

    The RDA for vitamin C is generous and, like other vitamins, doesn’t need to be met every day. The average intake from food alone is above the RDA in men, women, and children. In addition, many U.S. adults take vitamin C supplements. Deficiencies occur mainly in people who eat little or no fruits or vegetables—like the sailors who went to sea and got scurvy (Chap. 1). Such a diet is most common in alcoholics and older men who live alone.

    Although vitamin C is important in functions like wound healing and immunity, amounts in excess of the RDA do not enhance these functions. Enough is simply enough. The idea that taking even more provides additional protection isn’t supported by controlled studies.**

    *Collagen has hydroxyproline, an unusual amino acid made by adding an -OH (hydroxy) to the proline that’s already part of collagen’s amino acid chain. (Proline, but not hydroxyproline, is 1 of the 20 amino acids needed to make protein.) Vitamin C is needed to convert proline to hydroxyproline, so a deficiency interferes with the formation and maintenance of bone and connective tissue.
    **It’s been suggested that megadoses of vitamin C (1 gram or more—more than 16 times the RDA) prevent the common cold and cures cancer, but this hasn’t been found in double-blind studies (which adjust for placebo effects; Chap. 2). The only evidence for benefit is that huge doses may slightly lessen the severity of a cold, but even this isn’t a consistent finding in controlled studies.

    Phosphorus

    About 85% of the body’s phosphorus (P) is in bone, mainly as hydroxyapatite [Ca10(PO4)6(OH)2]. The other 15% is found throughout the body in the lecithin that makes up our cell membranes (Chap. 5, 9) and in DNA, RNA, ATP, etc.

    Phosphorus is widespread in food. Meat, milk, fish, grains, and nuts are rich sources. Even diet soft drinks have a fair amount. A deficiency is unlikely under ordinary circumstances, but has been seen in premature babies fed only breast milk. Breast milk has enough phosphorus for full-term babies, but not enough for premature ones. Thus, premature babies can need supplemental phosphorus for proper bone mineralization.

    Phosphorus deficiency can also occur with long- term use of antacids made of aluminum hydroxide, which can combine with dietary phosphorus and prevent its absorption from the intestine. Phosphorus deficiency can cause bone loss, weakness, and pain.

    Osteoporosis

    Osteoporosis literally means porous bones. Bone is normally porous (styrofoam-like), but the pores get bigger (become lace-like) as minerals are lost. In aging, the amount of collagen in bone also falls, making it more brittle. Bones can become so porous and brittle (especially in the wrist, spine, and hip) that they are easily fractured.

    A key factor in developing osteoporosis is one’s peak bone mass—the density (calcium content) of bone in early adulthood, when bone mass reaches its peak. The higher the peak density, the less likely osteoporosis will occur later.

    Osteoporosis is especially common in White, thin, older women. After age 50, wrist fractures are more common, usually from extending the arm to break a fall. Wrist fractures aren’t usually serious, but can indicate underlying osteoporosis.

    Fractures of the spinal vertebrae, which tend to occur after about age 55, are often painless, although some people suffer severe pain and disability. Outward signs of these fractures are a loss of height and the stoop of old age—often called dowager’s hump (see Figure 14.5).

    Fractures of the hip are the most serious and the most common fracture in women over age 70. Many die from complications (e.g., pneumonia), or are so disabled that they can no longer live independently.

    Although osteoporosis can be detected to some extent by losses in height, it’s usually brought to medical attention by a fracture. A puzzling aspect of osteoporosis is that the amount of bone loss isn’t necessarily related to the amount of pain or disability. One person may have compression fractures of the vertebrae and severe loss of bone density yet not suffer pain or disability, while someone else may have much less bone loss and experience debilitating pain.

    Osteoporosis has been hard to study because it’s detectable by imaging only after much mineral loss. Most studies of changes in bone mass compare people of different ages within a population (cross-sectional study) rather than following the same people as they age (longitudinal study). Longitudinal studies give better information (Chap. 2).

    Another complicating factor is that bone is not lost evenly throughout the skeleton or even throughout a single bone. Certain bones are often studied because of convenience, cost, etc. The wrist may be easier to study than the spine, but what one sees in the wrist may not reflect what’s happening in the spine or hip.

    How osteoporosis develops is an active area of research, and there are now more sophisticated methods of measuring bone loss, e.g., dual-energy X-ray absorptiometry.

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    Figure 14.4: Bone Loss with Age

    Risk factors

    Looking at all the ads for calcium supplements, you would think that osteoporosis was simply caused by a low-calcium diet. In fact, osteoporosis is a very complex disease brought about by an interplay of many factors. Some of these are:

    • Gender: Women are at higher risk. They have smaller bones, lower peak bone mass, accelerated bone loss at menopause, and live longer. Also, bone density increases most during fast growth, and growth slows for girls in their early teens, whereas it doesn’t slow for boys until their late teens. In other words, boys have more years of rapidly increasing bone density.
      • Both the male hormone testosterone and the female hormone estrogen help preserve bone mass. There’s a rapid fall in estrogen for women at menopause. Aging men have a small and gradual fall in testosterone and a rise in estrogen (which may contribute to their “mellowing” as they age).
    • Family history: Genes affect bone density and bone size and thus affect risk. Having a parent, grandmother, etc., with osteoporosis raises risk. However, it may not all be genetics. Families also share food and exercise habits.
    • Race: Osteoporosis is more common in Whites and Asians than in Blacks, presumably because of racial differences in bone density. Blacks have denser bones (even though their calcium intake is lower), and Black women have about half as many hip fractures as White women. Those with very light skin (e.g., those with naturally blond or red hair) also have higher risk. There aren’t as much data on osteoporosis in racial subgroups (e.g., Hispanics), but more is being gathered.
    • Physical activity: Normal physical activity preserves bone mass; prolonged inactivity causes bone loss. The importance of stress on bone is most dramatically demonstrated by the rapid bone loss during space travel, when astronauts escape the pull of gravity.
      • The greatest effect on strengthening bone mass occurs in going from minimal to moderate physical activity. Stress-bearing exercise (e.g., running) offers additional help in strengthening bone. But as with muscles, the effects are very specific, e.g., bone density is higher in the playing arm of a tennis player than in the other arm.
      • We sometimes need to be reminded that getting exercise doesn’t necessarily mean we have to take up a sport or work out. Vigorously scrubbing the bathroom or using a manual lawn mower is also good exercise. Also, note that women who engage in strenuous exercise to the point of amenorrhea (stoppage of menstruation) can lose bone mass (remember moderation). Amenorrhea has some of the same characteristics as menopause.
    • Body weight: Being overweight lessens risk of osteoporosis. The extra weight puts more stress on bones, thereby protecting bone density. The extra body fat provides more cushioning for a bone in a fall. (For those anxious to find health benefits of being overweight, another is less risk of tuberculosis—another is you live longer in a famine.)
      • Also, fat cells make some estrogen and become an important source when ovary-produced estrogen falls at menopause. Extreme thinness to the point of amenorrhea in premenopausal women (many ballerinas and women with anorexia nervosa have this problem) can cause bone loss because of insufficient estrogen.
    • Smoking seems to raise the risk of osteoporosis for both men and women. The reason isn’t entirely clear, but part of the reason for women is an earlier menopause, resulting in accelerated bone loss at an earlier age. Also, smokers tend to weigh less. As noted before, a heavier body weight and more fat tissue offer some protection.
    • Alcohol: Osteoporosis is more common among alcoholics. They tend to have poor diets, and alcohol can also lessen nutrient absorption by damaging the intestinal lining. Also, being tipsy raises the risk of falling—and fractures.
    • Dietary calcium: It’s important, of course, to meet your dietary requirement. As said earlier, most people over age 11 don’t meet their requirement. Ads encouraging more calcium in the diet are mostly directed at women, but adolescents should be the main focus. Adequate calcium intake during childhood and early adulthood is crucial for attaining high peak bone mass.
    • Other nutrients can potentially affect risk of osteoporosis, mainly by affecting calcium absorption and loss. Protein taken in high doses as a dietary supplement increases the loss of calcium in the urine. Much less is lost when the protein is from food. One reason could be that foods high in protein also tend to be high in phosphorus, which seems to help prevent the loss of calcium.
      • Lest one think that excessive phosphorus is helpful, a high intake of phosphorus reduces calcium absorption in the intestine. So, it seems that there’s neither loss nor gain in bone calcium when a diet is high in phosphorus and adequate in calcium. These conclusions are based on studies of adults, and there’s some concern that excessive phosphorus may have a detrimental effect on bone growth in growing children.
      • This is of particular concern for growing girls, since their diets tend to be high in phosphorus and low in calcium. Studies in young animals show that a diet that’s both high in phosphorus and low in calcium promotes bone loss. An important group to study in this regard are growing girls who are thin, small-boned, sedentary, fair-skinned, continually dieting, and who consume very little milk or milk products.
      • A high-sodium diet combined with a low-calcium diet can also contribute to osteoporosis. Excess sodium is excreted in the urine, and the sodium pulls some calcium out with it, making a calcium deficiency worse.

    Prevention

    As can be seen from the risk factors, the best that children and adults can do is eat a good diet (Chap. 4), get a moderate amount of exercise, and not smoke. The aim is to achieve high peak bone mass, and to slow bone loss during aging.

    Many new preventive therapies have been evaluated to retard bone loss in postmenopausal women. Estrogen-replacement therapy starting at menopause is effective; it reduces the number of hip and wrist fractures by about half.

    As with all medications, possible side effects of estrogen-replacement therapy is a concern, a major one being estrogen-related cancers. Estrogen used to be given alone (unopposed estrogen), and this raised the risk of cancer in the lining of the uterus (endometrial cancer).* Now, estrogen is given together with another hormone (progestogen) that greatly lessens—and possibly eliminates—the higher risk of this cancer.

    Estrogen therapy lasting more than 10 years seems to raise the risk of breast cancer. Studies have been inconsistent, as is common in studies looking for small effects in differing circumstances.

    A problem is that only recently have women been randomly assigned to control and experimental groups. In most studies, postmenopausal women who chose to take hormones have been compared with those who didn’t take them, and the two groups may differ in many other ways (Chap. 2).

    The best a woman can do upon approaching menopause is to discuss the available therapies with her physician in terms of her own risks and the potential benefits and risk of various therapies. Not all women are suitable for menopausal hormone therapy, including those who are many years past menopause and have already lost a lot of bone, those who have had breast cancer, and those with certain health problems such as liver disease or very high blood pressure.

    Several drugs that maximize the benefits and minimize the risks of hormone replacement therapy (e.g., lower the risk of both osteoporosis and breast cancer) are being developed. Non-hormonal drugs such as slow-release fluoride and bisphosphonates (e.g., etidronate, alendronate) are also used to prevent and treat osteoporosis.

    Fluoride hardens bone, much like it hardens tooth enamel. Bisphosphonates reduce bone loss. Also, don’t overlook common sense in preventing the falls that can fracture bone, e.g., installing handrails, strengthening muscles, and wearing eyeglasses to improve vision.

    *Postmenopausal obese women have a higher risk of endometrial cancer. Their extra body fat results in more estrogen made there—unopposed estrogen. Studies are being done now to see if giving progestogen to obese women after menopause will lower their risk of this cancer.


    This page titled 14.2: Bone is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Judi S. Morrill via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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