On the morning of 13 September 2017, Martha Myers was being prepared for surgery at north London’s Royal Free hospital. Stayin’ Alive was playing in the background. The anaesthetist had asked what she would like to hear, and she’d requested the Bee Gees – calming, “happy music” that took her back to her youth in Colombia. Meanwhile, 130 miles north, in Nottingham City hospital, Ryan Mace grappled with a pair of green surgical stockings, worrying that his gown would reveal his bottom as he walked to theatre. He’d wanted to do something like this since he was a teenager and now that the moment was here, he was ready. Back in London, south of the river at St George’s hospital, Steve Abbott had woken up feeling anxious. But his was a good ward to be on, full of high-spirited football chat that kept his mind off the day ahead. When the time for the operation finally came, he squeezed his son Ben’s hand before being wheeled away.

Martha, Ryan and Steve were in perfect health. But over the course of the morning, surgeons at the three hospitals tussled gently with fatty tissue and chopped through blood vessels to remove a kidney from each of them. The organs were packed in ice and whisked down corridors, then handed to couriers eager to get on the road.

Later that day, the theatre staff were back at work. Martha’s kidney had arrived in Nottingham, where it was transplanted into Ryan’s best friend, Lee Bennett. Ryan’s kidney went down the M1 to south London, and was given to Steve’s son Ben. And Steve’s kidney made the short journey across the Thames to be stitched into Martha’s husband, David. The circle was complete.

This is the strange magic of kidney donation chains. If you need a transplant in the UK, you can join 5,000 other people on the national waiting list for a kidney from a deceased donor. Or, if you’re lucky, a partner, relative or friend might offer you one as a living donor (for many people, as long as they have been fully assessed, it is possible to live healthily with only one kidney). But if you and they are incompatible by blood group or tissue type, a transplant may not be possible.

Instead, you and your would-be donor can join the UK Living Kidney Sharing Scheme (pdf). Four times a year, in what’s known as a matching run, a sophisticated algorithm works out chains linking incompatible pairs such as Martha and David, Ryan and Lee, and Steve and Ben. Reshuffled into compatible pairs, everyone in the chain who needs one ends up getting a kidney – not from their friend or loved one, but from a stranger.

The process relies on logistical masterminding by staff at the UK’s 23 transplant centres, as much as it does on the mind-boggling computer science it’s built around. Ideally, all the operations in a chain are scheduled to take place on the same day, within eight weeks of a matching run, and only after everyone has been painstakingly assessed to ensure the matches will work and they are well enough for surgery.

It’s a curiously under-recognised scheme, among the general public at least, but its impact is growing every year. The first UK kidney exchange took place in 2007, with a two-way swap between four people. In 2017/18, 127 of the 1,010 living kidney transplants carried out – one in eight – came about as a result of the sharing scheme. It’s also a British success story: the UK far outstrips any other country in Europe in this field, carrying out about half of all the transplants made possible by sharing schemes.

* * *

Just over a year after their surgery, five of the six members of one chain have agreed to meet to talk about their experiences and be photographed for the Guardian. (The sixth, Steve, lives in the US and joins us via FaceTime; later he will be digitally added to the magazine’s group shots.) Though some patients and donors matched in chains end up exchanging cards or letters after the transplants are complete, only a tiny proportion ever meet: anonymity is an important part of the process.

“Nice bunch of good-looking people!” says Martha as she and David come into the photo studio, where Lee and Ryan are waiting with their partners. As the writer who arranged this, I had been apprehensive about the meeting, fearing it might be awkward. But when I introduce 43-year-old Lee to Martha, 60, as the man who got her kidney, she grips him by the hand, pulls him to her in an embrace and begins to sob softly. “Nice to meet you,” he says, a broad smile on his face. “I’m looking after it for you.” They get chatting. Was he ever on dialysis, Martha asks: she supported her husband through nearly nine years of it before his first transplant in 1999; when that kidney began to fail, she was determined to donate one herself, rather than see him so ill again. No, Lee says, but he was on the brink: “You came up trumps just in time.”

Ben slips into the room 10 minutes later, all but unnoticed as the group exchange recovery stories and praise for the NHS. He and Ryan, his donor, greet each other with a warm blokeishness, a firm handshake and an exchanged, “All right?”

There’s an easy amiability in the room. David, 70, is an amateur expert on all things kidney-related and quickly established as the unofficial elder statesman of the group, while Ben, 27, is affable, dry-witted and hungover. Their conversations reveal a shared understanding of ill health and the joy that successful treatment can bring.

“With my first transplant,” David announces to the group, minutes after arriving, “I knew it was from a woman because the minute I started peeing I was sitting down on the toilet.” They talk about the thing they all remember about the day of the surgery: one of the donors had misunderstood the instructions to drink only clear fluids and enjoyed a cup of tea with milk in the morning, causing an unnerving last-minute delay. “I was literally yards away from the operating table,” Ryan remembers. “All I could think about was Lee.”

The culprit, it’s soon revealed, was Martha, winningly irrepressible whether she’s holding forth on the need for greater kindness in the world or sniffing back tears. “I love her,” Lee’s wife Karen says to me at one point. “I want to take her home.”

Away from the bustle of tea being made and the photographer arranging his shots, Lee and Ryan sit side by side, legs crossed identically over their knees. They have known each other only five years, but consider themselves best friends. Both grew up in villages near Nottingham, and later worked out that they had played rugby against each other at school. “We’ve got the same silly sense of humour, similar interests. We just hit it off,” Ryan, 42, says. “He’s such a relaxed person – I’ve learned a lot from him.”

But in the time they’d known each other, Lee, who runs a kitchen design business, had been getting increasingly ill. At the age of 28, soon after returning from honeymoon, he discovered he had polycystic kidneys, an inherited condition that his mother, who died when he was 18 of a brain tumour, had suffered from, too. Karen would have given him a kidney if she could, but already knew she had only one good one herself. Ryan, who works for the Royal Air Force in recruitment and selection, was quick to step in. “For me, it was a no-brainer,” he says. “The experts told me I was going to be fine, so why wouldn’t I try to help somebody? Seeing how Lee’s health was deteriorating, there was never a question in my mind.”

He cried after seeing Lee for the first time post-surgery. “It was just relief, I think,” he says. “The big thing for me was his eyes. As we’d got closer to the date of the operation, they were quite dark and sunken. When I saw him afterwards, it was like a different guy.” He has kept his green surgical socks; looking at them makes him smile. Lee still finds it hard to grasp that a friend would do what Ryan did. “It’s taken me back to being twentysomething again,” he says. “It feels like a miracle cure.”

* * *

The main business of the kidney is to remove toxins and excess fluids from the blood and turn the waste into urine; when they don’t, those things remain in the body, causing symptoms including high blood pressure, extreme tiredness and persistent headaches. Left untreated, kidney failure is eventually fatal. Aside from a transplant, dialysis – which filters the blood using either a machine or the inside lining of the abdomen – is the main treatment; but it only partially compensates for lost kidney function, is disruptive to daily life and can cause unpleasant side-effects. Transplants save and transform lives.

“Kidney disease and dialysis affect everything that we would take for granted in life,” explains Lisa Burnapp, NHS Blood and Transplant’s lead nurse for living donation. “They can impact on your work, your ability to have children, what you can eat and drink, where you go on holiday.” And while the average wait for a kidney from a deceased donor has fallen to just over two years, a living kidney is preferable – because it comes from a healthy person and can be transplanted in a planned fashion. It’s often described as the Rolls-Royce of transplants.

I joined Burnapp at Guy’s hospital in London on a sunny morning earlier this year, as she looked through the results of a matching run carried out the day before. The algorithm, developed by Glasgow University’s school of computing science, identifies several different types of swap. There are circular exchanges between either two or three pairs, but also chains started by altruistic donors – people who simply want to donate a kidney to anyone who needs it, rather than for someone close to them. These are gamechangers, Burnapp told me, because one of them can enable up to three transplants to take place, with one kidney ending up with a patient on the main waiting list.

Altruistic donors on the register used to be given the chance to opt into the matching runs. But from this year, in a bid to increase transplant numbers, they have been added automatically – unless there’s a high-priority, difficult-to-match patient on the waiting list for whom they’re a “golden ticket”.

We peered at Burnapp’s laptop: the matching run had been triggered at 10.39am plus 10 seconds. One hour, 32 minutes and 32 seconds later, the algorithm had sifted through 271 patients and their pairs to identify 82 transplants, including 17 three-way exchanges. It’s the highest number of transplants yet to be identified in a run. Exciting, said Burnapp, and proof that efforts to develop the scheme are working. “To me it’s really heart-lifting, because these are patients who would never see a transplant. It just means everything, really.”

Since the living kidney scheme started in 2007, about 2,000 patients have registered on it, and more than 900 people have benefited from a transplant – including 19 children, the youngest aged three.

As well as meeting a chain who have had surgery, I want to speak to a pair who are currently going through the process. I meet Richard and Maureen Ventre at the Royal Liverpool hospital with their donor coordinator, Ann Strong, 12 days before Maureen is due to give a kidney that will mean Richard gets a much-needed transplant. The retired managing director of a family construction business, he was diagnosed with a collapsed kidney in 2016. It was a shock: unlike a lot of kidney disease, his problems came on quickly. “It’s like you’ve got a nice lively battery in you and all of a sudden it’s just gone,” he says.

He and Maureen met in the heatwave of 1976, working a summer season in a grand Georgian hotel in Torquay. He was a 23-year-old student from Liverpool and she, aged 17, had come all the way from Glasgow. The scousers, geordies and Glaswegians always seemed to gravitate towards each other, they remember, and one night they found themselves out on a double date. Today, they have the calm, quiet closeness of a couple who have been married for almost four decades.

Six years ago the Ventres lost the eldest of their four children, Tim; he drowned in a pool in El Salvador at the age of 30. He was a traveller, Richard explains, and they would love to follow in his footsteps; south-east Asia, maybe, off the beaten track. They aren’t sure that will be possible now, but they keep it in mind.

Maureen, who worked until recently in admissions at a local college, is twinkly-eyed and chatty, but when I ask about her decision to donate, her face crumples up with tears and no words come out. “It’s just obvious,” she eventually mutters. “No, it wasn’t hard.”

In fact, she and Richard, 59 and 65 respectively, are compatible, so her kidney could have gone straight to him; but they decided to use the sharing scheme in the hope of getting him an even better match. This is encouraged: the more pairs in the pool, the more transplants are likely. They are now in a long altruistic (and anonymous) donor chain: Richard’s new kidney will come from the altruistic donor and Maureen’s will go to a patient in another pair, whose donor’s kidney will then go to someone on the waiting list.

At 8am on the dot on the morning of their surgery, Strong gets a call on her mobile from the hospital Richard’s kidney is coming from; everything looks OK there. She checks in with the centre that will receive Maureen’s kidney; it’s also good to go.

I join her in theatre later. The kidney must be cut and cauterised away from the yellowy fat that surrounds it until only the vein, arteries and ureter are holding it in place; the persistent snipping of the cutting tool rings out in the cool hush of the room. The keyhole surgery is done via three 1cm incisions, the surgeon, Ajay Sharma, wielding the instruments while the registrar holds the camera (the kidney will come out of a larger cut, around 7cm long). They stand almost shoulder to shoulder, never taking their eyes off the large screen above them, the pink of the tangle of kidney and fat it shows reflected in their glasses.

Dan Ridgway, the second surgeon, watches from the sidelines. “These are the ones that keep me awake at night because, essentially, you’re a healthy person having an operation you don’t need,” he says. “I don’t even call them patients. They’re VIPs.” In the afternoon, he’ll be the one fixing Richard Ventre’s new kidney into place.

Once the arteries are clamped and cut, it’s crucial for the health of the kidney that it’s removed as soon as possible. The time each is done is recorded on a whiteboard. “Time on the board, artery one,” Ridgway says. “12.06,” comes the instruction from Strong to a nurse. At 12.09, they repeat the process for the second artery. The vein is cut last, and suddenly, at 12.10, a bowl containing the kidney is thrust towards Ridgway. His task now is to swiftly cool it and flush out the blood. A tall man, he sits hunched over a trolley at the foot of the operating table, sets the kidney in a bed of sterile ice and uses tweezers to locate an artery. Hanging beside him is a bag of perfusing fluid – the liquid used to wash out the kidney – and once he has the artery between his thumb and forefinger, the registrar inserts a tube. The fluid rushes in and the water in the bowl gradually turns red as the blood flows out. The kidney gets paler. “It’s lovely to see it come to life when they transplant it,” Strong says, “because it goes in that colour and then pinks up.”

Ridgway skilfully peels away the remaining fat and clips it off with long, curved scissors. “A good-sized kidney,” he says, giving it a little squeeze.

The organ is triple-bagged, floating in perfusing fluid, and Strong buries it in an icebox. Ridgway gets a text from the surgeon who has removed the kidney that is going to Richard Ventre. “Everything fine,” it says. “Good luck.”

Down the neon-lit corridor, the courier is already waiting for us and, as soon as the paperwork is filled in, she’s off, back down in the lifts and towards the hospital where the next patient in the chain is waiting. It’s 12.31pm. You can get a kidney pretty much anywhere in the UK in four hours, Strong says. Couriers will often use blue lights, and the organs are flown between cities for longer journeys. “Every hour counts,” she says. “The longer it’s out of the body, the slower it might be to work.”

Just after 4.30pm, Ridgway – who’s been to Anfield to buy a Liverpool away kit for his son during the surgical break – pops his head into Strong’s office: the donated kidney has arrived. We fetch it from the ward and Strong hands him the box with a cheery, “It’s all yours.” He lopes into the lift, heading straight up to theatre. In the office, Strong reclines in her chair and beams. “There you go,” she says. “It’s great, isn’t it?”

* * *

When we meet, David tells me he hadn’t wanted his wife to donate a kidney. “I just didn’t want her to go through it,” he says. “But she made me realise I had to do it – not just for myself, but for her.” They got together six months after Martha arrived in England from the Colombian city of Bucaramanga to study business in 1985, and married five years later. “Three months down the line, he got ill and the best years of our lives for having a family were gone,” she says. By his last year on dialysis, David was so weak that he was forced to close his successful design and artwork consultancy, aged just 49. “What kind of life is that for the two of us?” Martha had asked of the prospect of her husband going back on dialysis. “I didn’t think that I could cope with more,” she says.

When he woke up from surgery last year, David couldn’t believe how well he felt. “I was sending texts, emails,” he says. “I felt great.” He listened to music on the radio a lot in the days that followed and frequently found himself in tears, overwhelmed by the realisation of what Martha, and the kidney exchange scheme, had made possible.

“Some people say, ‘You’re unlucky because you’ve been ill so long’,” he says. “I’m lucky. I’m lucky that Martha came into my life and I’m very lucky that I’ve got this kidney. I’ve got a chance for us to go and enjoy ourselves.”

This was Ben’s second transplant, too; he suffered kidney failure at the age of 19 and his mother donated directly to him three months later. Early in 2016, the recruitment consultant was told by his doctors at St Helier hospital, in Surrey, that that kidney might only last another six months. He professes not to have minded the two months he spent on dialysis before his first transplant (“I got to watch Come Dine With Me for four hours three times a week and eat really nice sandwiches”) but this time, with a partner, a demanding job and a football team full of teenagers to coach, he was desperate to avoid it.

“When we’re together, we’re like two peas in a pod,” Ben says of his father. The night before surgery they went to the pub to watch their beloved Chelsea football club beat Azerbaijan’s Qarabağ FK 6-0 (“He had a pint, I didn’t”). Several months earlier, they were due to be part of another chain, but the operations were cancelled the day before because one of the recipients was unwell. (For the run I looked at with Burnapp, 57 of the 82 transplants that the algorithm came up with – 70% – ended up happening; the figure is increasing and the completion target is 75% by 2020.)

The delay was difficult, but Ben was told that Ryan’s kidney – “a big juicy one” – was, in fact, a better match for him. “It’s working wonderfully,” he says. Of everyone in the chain, he’s the least given to dwelling on the emotional side of what he’s been through, though he does remember crying at his first dialysis session, struck by the reality of how ill he was.

I spoke to Ben’s dad, Steve, on the phone. Although he lived overseas and ended up travelling back and forth to the UK seven times in a year as a result, he was determined to donate. “He’s my son,” the 55-year-old builder said. “Simple as that. You don’t like to see your kids suffering.”

Steve hoped his kidney would give someone a good life but had never really thought about who might end up with it. “As long as Ben got his kidney and he was fine, you could have put it in a pie and cooked it, for all I cared,” he said. But when Ben gets hold of him on a video call at the photoshoot and David tells him how grateful he is, something shifts. “When I started speaking to David it sort of sunk in a bit,” he tells me later. “It was done for Ben, but I have helped someone else have a normal-ish life.”

David, meanwhile, is thrilled to discover that his kidney came from a fellow Chelsea fan. He emails Steve that evening, and later shares the message with me: “It is an incredible feeling to think that someone like you has made this incredible gift for a loved one and you have ended up helping not one but two people,” it says. “A million thanks to you.”

A few months after their surgery, I catch up with the Ventres. Richard’s kidney is working well and he’s started playing a bit of golf again. Maureen still feels lethargic at times, but she’s enjoying walking and swimming, and hopes to return to yoga soon. They are due to spend their 39th wedding anniversary at the hotel in Torquay where they met, their first trip back since the 70s. And they still hope to travel to some far-flung destinations one day, if the doctors approve.

Meanwhile, the members of our six-strong chain plan to keep in touch. Ryan and Ben have been chatting online about their recruitment work; David and Martha want to take Ben and his dad to a Chelsea match next time Steve is in the UK, maybe even get a box. Martha and Lee are meeting for a drink when he’s next in London, and she hopes to arrange a reunion for everyone a year from now.

When they met, Lee told the group he’d thought he’d only be interested in meeting Martha, his donor. “But having met everybody it’s strange how it all feels interlinked,” he said. “None of it would have happened if it hadn’t been for everyone.”

• For more information, go to organdonation.nhs.uk/about-donation/living-donation Hear Rachel Williams talk about the story behind this article on the new Guardian podcast, Today in Focus, later this month, at theguardian.com.

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