What are varicose veins? Veins are the blood vessels that carry blood back to the heart. Varicose veins are abnormally enlarged and tortuous veins that are visible just below the surface of the skin. Smaller veins in the skin itself are sometimes called "thread veins" or "spider veins". Although these may be unsightly they are not the same as varicose veins and rarely cause discomfort or complications. What causes varicose veins? Varicose veins are due to weaknesses in the wall of superficial veins leading to stretching. This causes failure of the one-way valves inside the veins. These valves normally only allow the blood to flow up the leg towards the heart. If the valves leak, then blood can flow back the wrong way on standing. This reverse flow causes increased pressure on the veins, which swell and become varicose. The increased pressure can cause aching and complications such as eczema, discolouration and leg ulceration. Varicose veins often run in the family and as you get older they are more likely to occur. They may also be caused by standing occupations, pregnancy or weight gain which increase pressure on the leg veins. How common are varicose veins? Approximately half the population has some form of venous disease, and varicose veins affect up to 30% of all adults. How is the operation performed? Immediately before your operation, the surgeon will mark the veins on your leg with you standing up. He may also scan your leg using ultrasound. The operation is usually performed under a general anaesthetic, but other methods are possible. The anaesthetist will visit and discuss the best options for you. The commonest operation is where a small cut is made in the groin or behind the knee over the top of one of the main varicose veins. This is then tied off where it meets the deeper veins. If possible, the main varicose vein on the inner aspect of the leg is then stripped out. Blood still flows up the leg along deeper, unaffected veins. The cut in the groin (or behind the knee) is closed with a dissolving stitch, hidden under the skin. The other veins marked before the operation are then pulled out through tiny cuts called microphlebectomies. These are often covered with adhesive strips, called steristrips. A dressing will be placed on the cut in the groin, and your leg will be bandaged up to the top of the thigh. The bandage normally remains in place for about five days and a fitted stocking is then worn during the day for two further weeks. How successful is varicose vein surgery? Many clinical studies have been published which document the success of varicose vein surgery. This is one of the most successful ways of obtaining a good and durable result. Like any medical treatment, however, it has certain risks, which will be explained to you. What should I do before the operation? If you are having your procedure under a general anaesthetic and are taking an oestrogen containing oral contraceptive pill, you should stop this six weeks before the treatment. If you require contraceptive cover during this period, you may wish to discuss changing to a progesterone only mini pill with your G.P. The progesterone only mini pill does not need to be stopped before surgery. Usual oral contraception may be recommenced in the week after surgery. Hormone replacement therapy (HRT) does not need to be stopped before surgery. On the day of surgery please do not apply any moisturiser to your legs. Ensure that you wear loose trousers or a skirt and loose shoes or sandals so that there is room for the dressings. What should I do after the procedure? Following the operation a small dressing will be placed over the groin wound and a compression bandage will be applied. This should stay in place for about five days. You will be given specific instructions about this. A supplied specially measured stocking should then be put on and worn during the day and removed at night for the next two weeks. You may have a shower 4 days after your operation, and on the 10th day you should have a bath to soak off the adhesive strips. You will be encouraged to mobilise straight after the procedure, but also to rest with your feet elevated so that your heels are higher than your hips to aid the drainage of excess fluid from the tissues and assist healing. Slight discomfort is normal. Local twinges of pain may occur in some patients. In the first week after the operation you may need to take a mild painkiller such as Paracetamol to relieve discomfort. Frequent (2 – 4 times daily) short (15 – 20 minutes) walks are beneficial for the first 7 – 10 days post- operatively. You will be encouraged to rapidly return to normal activities but it is advisable not to drive for one week after the operation. A short break from work (7 -10 days) is sensible depending on your occupation. Swimming, vigorous exercise and hot baths should be avoided for two to three weeks as this increases the risk of blood flow returning to the treated vein. It is wise to avoid long-haul flights for 6 weeks postoperatively. It is advisable to leave at least one month after your operation before a holiday or special occasion. What to expect during the recovery period Sometimes a little blood will ooze from the wounds during the first 12-24 hours. This will stop on its own. If necessary, press on the wound for ten minutes. When the bandage is removed there will be a variable amount of bruising, depending on the extent of your varicose veins preoperatively. This is quite normal and settles 2 - 3 weeks after the operation. It is very common to have some lumpy areas where the veins have been removed and this settles 6 – 12 weeks after the operation. Although the majority of healing is complete at 6 – 8 weeks it is usually 3 – 4 months before the final result is achieved. There will be a small scar in the groin, or behind the knee, which will fade over time. The other wounds on your legs will continue to fade for several months before disappearing. Are there any complications following this procedure? Some degree of temporary bruising is inevitable and this usually settles within 2 - 3 weeks. There is a risk of altered pigmentation or staining of the skin where the veins have been removed. This normally disappears within a few months but is permanent in 5% of patients. Infection in the wounds may occur (3%) which might require a course of antibiotics. Sometimes there is numbness around the wounds or ankle. This is unavoidable and is due to pulling on nerves during the operation affecting up to 10% of patients. It usually settles after some weeks or months, but may occasionally be permanent. Some people may develop worse or new “spider” or “thread” veins close to the sites of treatment. These can be treated by micro injection techniques if necessary. As with all surgery there is a risk of deep vein thrombosis (DVT), although this is rare (approximately 1 in 250). You will be supplied with compression stockings and encouraged to mobilize to prevent this. There is a risk of recurrence of varicose veins (3 to 6% per year), as you are clearly disposed to them. The taking of regular exercise, the avoidance of becoming overweight, and the wearing of light support tights or stockings will all help prevent you being troubled by varicose veins in the future. Your health professionals will make every effort to make your treatment as safe as possible. However, complications can happen with any medical treatment.
MATH 232 – Scientific Calculus II Homework Supplement S16 1. By writing out the first several terms (like we did with the example from class), try to derive an with k ≠ 0 , is known as the discrete exponential model (DEM). a. Use the analytical tool demonstrated in class (i.e. look for a pattern) to derive an explicit formula for the solution of the DEM, with initial conditi
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