What's the formula for the circumference of an ellipse? Last weekend, Gary E. Hartman, a 56-year-old surgeon at Children's Hospital, racked his brain but just couldn't dredge up the answer from high school geometry.

So he went online and found the formula, devised in 1609 by Johannes Kepler to describe the paths of heavenly bodies. On Tuesday afternoon, he put the simple equation to more down-to-earth uses. It helped him decide to delay the surgical separation of conjoined twins Jade and Erin Buckles by one week. The operation originally was scheduled for today.

Perhaps the biggest challenge of the complicated surgery will be closing the gaping wounds that will be left when the 15-week-old twins from Woodbridge are delicately cut apart. A month ago, the surgeons implanted four "tissue expanders" in their abdomens -- inflatable plastic reservoirs that have been slowly injected with water to stretch the overlying skin and make it grow. They are now bulging and elliptical -- but by Hartman's calculation, it will take one more week of filling them to grow the amount of skin he thinks he will need.

"We think it's going to be very tight," Hartman said. "If we can get three more injections between now and separation, we think that will give us between a half and a full centimeter [of extra skin] in each location."

Of course, he added a minute later, "that's just a guess" -- despite the precision of his calculation. How much skin there will be -- and how much he will need -- won't really be known for sure until the day of surgery. But if an extra week gives the Buckles twins a better chance on the most important day of their young lives, Hartman is sure the delay is worth it.

The decision to postpone -- reached in consultation with Michael Boyajian, the plastic surgeon at Children's who implanted the tissue expanders -- illustrates the degree to which the separation of Jade and Erin is a work in progress.

Already, virtually every department at the 279-bed hospital in Northwest Washington is doing something out of the ordinary to prepare for the surgery, now scheduled for June 19.

The capacity of the electrical service and air-conditioning system in the operating suites is being increased. An unprecedented amount of blood for a single procedure is being collected. The psychology and social work departments are helping the Buckles family -- parents Melissa and Kevin Buckles and two other children -- prepare for every conceivable outcome.

Several physicians have tested equipment in practice operations on piglets in the hospital's research labs. Three experts in quality improvement have written clinical "pathways" and decision trees that outline what should happen in nearly every eventuality -- good or bad -- the twins could face. In many cases, plans are being revised almost as soon as they are drawn up.

Although the original date of the operation was set nearly two months ago, Hartman and others planning it have worked hard to keep the enormous enterprise flexible enough to accommodate the unexpected.

"It's actually something that we talked about in some of the early planning meetings. There's this normal momentum to being on the OR [operating room] schedule that was even greater with this case. And we don't want the momentum of the process to push the clinical decision."

The skin growth issue took on added importance when it became clear that Erin's heart lies horizontally, not vertically, and that its tip will stick well out of her chest once she is separated from Jade.

"Two and one-half centimeters, just what the scans show," Hartman said Tuesday afternoon as he measured the tongue-shaped tip of Erin's heart as it projected nearly one inch from a life-size model of the twins that had arrived that day.

The medical team asked Medical Modeling of Golden, Colo., to help it visualize the three-dimensional relationship of the internal organs in the area where the babies are connected. Using data from CAT and MRI scans and employing a computerized "3-D printing" system, the company fabricated three models out of plaster powder. The most elaborate one shows the girls' hearts in different colors and opens at the expected plane of separation to demonstrate how big a hole will have to be closed or covered on each girl with skin and synthetic substitutes. The company has done about $10,000 worth of work, most of it donated, and expects to be paid no more than $3,000, its president said.

Hartman's calculation of the amount of skin Erin needs -- and might get from another week of tissue expansion -- is based on a plan to simply cover her heart as it lies. The surgeons won't try to push the tip back inside her chest, a procedure that could compress the heart's chambers and prevent them from filling properly. They also won't turn it vertically, which might kink an artery or vein.

They'll worry about moving it -- if ever -- another day.

"The heart's going to stay where it wants to stay," Hartman said.

The realm of Erin's heart, however, is about the only place where a laissez-faire view reigns.

Seven new electrical circuits are being lowered through the ceiling into Operating Room No. 2, allowing the number of electrical outlets to be roughly doubled to 52. That, in turn, will let the room accommodate two heart-lung machines, as well as various pieces of ultrasound imaging equipment not usually on hand.

The extra devices and large number of people will pour heat into the space, taxing the air-conditioning system, which now moves air at 2,100 cubic feet per minute. By the day of surgery, the capacity will be 3,600 cfm.

A week ago, Hartman and the hospital's chief of radiology, Dorothy Bulas, tried an experiment in the hospital's large-animal lab. They wanted to see whether the surgeons would be able to cut the shared liver with a "harmonic scalpel" -- which coagulates blood with ultrasound waves as it cuts -- at the same time a radiologist showed them the best route through the organ with ultrasound scans.

They tried it on a two-week-old pig. It didn't work. The scalpel caused too much interference with the ultrasound image.

"It's clear we're going to have to get an image, then have him cut, and then get another image. We're going to have to keep on showing him he's going in the right direction," Bulas said the next day, back in the lab trying out a variety of ultrasound transducers on a second animal.

In the blood bank, meanwhile, people were collecting an unprecedented amount of blood for the operation -- an effort made easier by the delay.

Naomi L.C. Luban, the physician who heads the blood bank, and her colleagues consulted six children's hospitals that altogether had separated eight pairs of twins. They also looked back at Children's own experience with complicated operations. They made "a bunch of different estimates of worst possible circumstances that might occur in the girls' procedure," Luban said.

They came up with this goal: 80 units of packed red blood cells; 10 to 20 units of whole blood; 24 units of platelets; 84 units of fresh frozen plasma; and 36 units of cryoprecipitate, a concentrate of specific clotting proteins. (A unit of whole blood is about a pint.)

That is four times the amount of blood ever stored for a single procedure at Children's. It is roughly 100 times the amount of blood now flowing through the twins' veins. If there is severe bleeding during the operation, there will be enough blood. If the children need to go on heart-lung machines, there will be enough blood. If the girls get low on platelets, which slow bleeding by forming the initial plugs in severed veins and arteries, or on coagulation proteins, which weave fibrous clots to permanently seal the vessels, there will be plenty of both.

The blood bank is getting ready for one of the worst complications of major surgery, "dilutional coagulopathy," when virtually all the body's blood is lost and clotting goes haywire. If that happens, everything -- red blood cells, platelets and all the components of the clotting "cascade" -- must be supplied from the outside. It's a catastrophe that is rarely overcome unless doctors have planned for it.

At one of the staffwide meetings to plan the separation, the hospital's chief of surgery, Kurt Newman, emphasized how important it was for everyone to make provisions for bad outcomes. Among his questions that day:

Do people know what they will do -- and how they will help the Buckles family -- if one baby dies in the operating room? What will happen if the twins, now so similar that only the parents can reliably tell them apart, have drastically different outcomes? What happens if one is permanently disabled?

"We're expecting everything to go well. We're trying for everything to go well. But certainly, taking care of sick kids, we all know that one or both may not do well," Newman told the crowded room. "I just want to be sure that we're prepared for that."

Some things, though, are proceeding exactly as anticipated.

More than a month ago, Hartman was asked how he thought the Buckles case would affect him, compared with other difficult ones in his 25-year career. He answered confidently: "This is the kind of case I would stew upon for the week before it happens. The night before, I would not sleep well. And I would be relieved when it was over."

That process, he said last week, was right on track.