For years, Grace Silva had experienced odd episodes with her throat — bouts of swelling and radiating pain that seemed to resolve with antibiotics — but her doctors couldn’t explain what was wrong. Finally, after a flare-up in the summer of 2010, Grace was referred to a specialist, an ear doctor who felt something amiss on the left side of her throat: a lump. The Silva family agreed that it was time to get Grace, then 54, to a thyroid specialist. Grace’s daughter Melanie tracked down the name of one at Brigham and Women’s Hospital, a 90-minute drive from Grace’s brown clapboard split-level near New Bedford, Mass. In September 2010, the specialist delivered the diagnosis: anaplastic thyroid cancer. It was bad, he warned her, and she would need surgery. Grace’s other daughter, Karrie, was marrying in a few weeks. “Can’t it wait?” Grace asked. It could not. “And whatever you do,” the specialist said, “please don’t look it up on the Internet.”

Medical texts describe the prognosis for anaplastic thyroid cancer as “poor,” but that hardly captures it. If every cancer has a personality, this one is notoriously aggressive. Its tendrils of tissue are so invasive that by the time of diagnosis, it is often too late to operate safely. Radiation or chemotherapy rarely buy much time, and even when all traces of the tumor are eliminated, it usually reappears. Anaplastic thyroid tumors are also known for their aberrant firmness, more akin to wood than flesh. As they bloom, the tumors can tighten like a noose, constricting the windpipe and giving their victims a sensation of perpetual drowning. This panicky “air hunger” can be mitigated with escalating doses of morphine, but it’s a miserable, desperate end that, once witnessed, is not easily forgotten. The oncologist Grace was sent to, Dr. Jochen Lorch of the nearby Dana-Farber Cancer Institute, had watched a patient die this way during medical school, and it filled him with such horror, such helplessness, that for years he felt sure he would never pursue a career in cancer treatment.

Of course nobody dreamed of saying these things to Grace at the time. A surgeon removed her tumor, and she was able to attend her daughter’s wedding. She then withstood a long stretch of radiation, every weekday for more than a month — vomiting into a bag as her husband, Joe, did his best behind the wheel of their brown Chrysler. Her doctors knew they had to hit the cancer with everything they had. By the end of her treatment in December 2010, Grace, a once-vivacious Portuguese woman with dark eyes and raven hair, had lost more than 30 pounds and could barely eat or talk.

In March 2011, Joe drove Grace back to Dana­Farber. She had started to recover some of her strength, and the day had come to learn the results of her first follow­up scan. The news wasn’t good. Less than three months since her last radiation treatment, the cancer had already spread to her lungs. The largest mass, on her right lung, was more than an inch in diameter. This is the way with anaplastic thyroid cancer.

Lorch sat with Grace and Joe in a cream-colored exam room, a red biohazard box nestled under one counter, and explained that all the standard treatments had been exhausted. He told them about an experimental trial for aggressive thyroid cancers that hadn’t responded to standard treatments, and Grace agreed to enroll. The drug, everolimus, was used in transplant surgery to prevent rejection, and it had been approved for some use in cancer. Lorch had seen indications that the drug could work in the thyroid, but he didn’t have high hopes for the anaplastic cases — its long track record had been too dismal. “Partly we were motivated,” Lorch told me, “by the fact that we didn’t have anything else.”

In the days after starting the trial, Grace found herself standing in front of the mirror in her bedroom, taking in her diminished reflection, beseeching God to give her some kind of sign. If her time had come, she wanted to know, for the sake of her husband and three children. “If I’m going to die, I need time to prepare,” she recalled thinking at the time. Either way, she felt sure that God would hold her by her “right hand,” as in one of her most cherished lines from the book of Isaiah.

Two months later, in May, Joe drove Grace to Dana-Farber for her follow-up scan. Lying motionless as the CT scanner began its inquiries, she thought of Isaiah and prayed. If the everolimus had failed to slow the cancer’s advance, it would be time to begin the work of getting her affairs in order. But the scan results, plain to even an untrained eye, were shocking: The largest mass had shrunk to half its previous size. Everywhere there were signs of retreat. Lorch said he had never seen such a rout. All six of the other anaplastic thyroid patients on the trial eventually died, but Grace’s tumors shrank until they barely registered.

Grace’s case became the subject of intense scientific scrutiny. How could such a notoriously recalcitrant cancer simply collapse? Why had she alone responded so extremely? Nobody was claiming that she was cured. But by the end of 2011, Grace felt this much was sure: Having asked for a sign, she had become a walking miracle.

What happened to Grace is sometimes called by another biblical name: the Lazarus effect, after the story in which Jesus stands outside the tomb of Lazarus of Bethany and summons him back to life. Many veteran oncologists have seen cases like Grace’s, and the stories of these unlikely recoveries, shared online or by word of mouth, have become a source of hope for patients. Yet for the field itself, the Lazarus effect has been a source of persistent frustration. In 2011, for example, the Food and Drug Administration withdrew its support for the treatment of breast cancer with Avastin, a drug with proven efficacy on tumors in other organs. Some breast-cancer patients had experienced powerful responses — and owed their lives to the drug — but most patients weren’t helped and were instead exposed to unnecessary side effects. With no way to predict the results, the drug was as good as useless.

Today patients like Grace have come to be known as exceptional responders, and cancer researchers have finally begun to unravel the puzzles they pose. In a cancer, some of the body’s cells develop genetic aberrations, growing and spreading uncontrollably, and there are myriad variations on this theme. While physicians recognize hundreds of types of the disease, genetic analyses suggest that the true number is far higher. The closer that scientists look at tumors, the more mutations they find, to the point where it may be impossible to count the types of cancer. Really, every patient suffers her own personal cancer, and when a drug is perfectly aligned to it — to the exact set of mutations driving the tumor — the result is an exceptional response. In such a case, if scientists could catalog the tumor’s mutations, they would have a shot at reconstructing a play-by-play — how the conflagration began, how the drug smothered it — and, from this, gain insights that could help others.

The power of this approach was first demonstrated a few years ago, at Memorial Sloan Kettering Cancer Center in Manhattan. In April 2009, Sharon K., who was 68 at the time, had been told by her local doctor that her bladder cancer had morphed into a muscle-invasive form: It had become aggressive and difficult to contain. At Sloan Kettering, she was given chemotherapy, followed by a cystectomy, which involved removing the bladder and fashioning a new one out of a portion of small intestine. “I felt like my insides were going to fall out,” said Sharon, who asked that her last name not be used to protect her privacy. Still, a few months later, the cancer returned.

In February 2010, running out of alternatives, Sharon joined a clinical trial at the center, with instructions to take two pills every morning and return for regular checkups. Thousands of trials are open in the United States on any given day, and for people like Sharon, who traveled from Florida to take part, they are an opportunity to take advantage of the latest scientific ideas. But the odds are generally long: Historically, less than 7 percent of cancer drugs tested in humans eventually win F.D.A. approval.

At Sharon’s first follow-up scan, the tumors were in recession; within months, they were gone. Her doctors were thrilled. And yet the trial Sharon had joined was a failure. Of the 44 other patients, just one saw his tumor shrink in a meaningful way. Dr. David Solit, a researcher at Sloan Kettering, joined a meeting with his colleagues there to discuss the trial’s results, and he remembers the feeling in the room, familiar to anyone in the field. “O.K., we’ve had no new effective bladder cancer treatments for 30 years, and we did yet another clinical trial that was [based on] a reasonable idea,” he recalled. “This is a disappointment. Now let’s give up and move on to the next thing.”

But before they did, they wanted to look into Sharon’s exceptional response. In the previous few years, a new technique called next-generation sequencing had made reading an entire genetic code exponentially faster and cheaper. Curious to see what the technology was capable of, Solit and his colleagues sent Sharon’s tissue to Illumina, a sequencing company with headquarters in California. Three months later, Illumina sent back what amounted to two Human Genome Projects: a complete readout of her DNA, totaling some three billion base pairs of code, and then another, equally large, for her tumor. After months of investigation, considering the potential significance of the tumor’s various mutations, the researchers settled on a prime suspect, a gene called TSC1. When they re-examined the failed clinical trial, they discovered that the bladder-cancer patients genetically similar to Sharon had done noticeably better, staying in the trial substantially longer. The problem hadn’t been the drug, but knowing exactly who should receive it.

When Sharon’s doctors published a paper on their findings in the journal Science, researchers at the National Cancer Institute in Bethesda, Md., immediately understood the broader implications. They sifted through a decade of “failed” clinical trials, thousands of cases, and found that more than 100 patients had experienced impressive positive effects. More Sharons were out there, each potentially harboring a secret about how to defeat cancer.

In 2014, the institute started the Exceptional Responders Initiative, and since then the case reports have come in, each a tantalizing mystery. What explains the patient with a Stage 4 esophageal cancer that spread to the liver but then disappeared three years ago? Or the Stage 4 adenocarcinoma patient who experienced a complete remission? What can we learn, from each Lazarus, about how to save the lives of others?

The first surviving record of treating cancer dates to around 1600 B.C., in an ancient Egyptian papyrus: tumors of the breast, excised and cauterized with what is described as a “fire drill.” In the centuries since, oncology has retained something of this elemental character. It is a bodily assault, brutal but necessary, guided largely by trial and error. The Boston hospital where Grace was treated takes its name from the cancer researcher Sidney Farber, who pioneered a treatment for childhood leukemia in the 1940s using aminopterin, a poison that racked his young charges but held the disease at bay, at least for a time. What works and what does not with one generation of patients is used to guide treatment for the next.

This idea, more broadly, has been called precision medicine: the hope that doctors will be able to come to a far more exact understanding of each patient’s disease, informed by genetics, and treat it accordingly. It is in cancer where this has advanced the furthest, and the exceptional responders provide a glimpse of what precision medicine might mean. When Grace’s tumor was sequenced, scientists found a mutation in TSC2, a sister gene to TSC1, the one mutated in Sharon’s cancer. Sharon, like Grace, had responded to everolimus, and so the genetic similarity suggested that their cases were not flukes, that their seemingly different cancers shared a deep connection. In this sense, Grace’s anaplastic thyroid cancer more closely resembled Sharon’s bladder cancer than other thyroid cancers. A more precise oncology would have assigned both to everolimus on purpose, not by chance.

Still, the two women’s tumors were not identical: While Sharon’s cancer vanished, Grace’s still lingered, even if it was harder to discern. Grace’s disease barely showed on the scans, and she looked healthy; friends and family found themselves forgetting, from time to time, how seriously ill she was. Dr. Nikhil Wagle, a Dana-Farber physician and scientist who worked on the genetic analysis of her tumor, was once involved in the case of a 38-year-old man with metastatic melanoma. A photo, taken when he joined a trial for the drug vemurafenib, shows the man’s chest, pale with a greenish cast, covered in large, oblong tumorous bumps. Cancer derives from the Greek for “crab,” and in this case, it was easy to see how the ancients imagined crab claws in cancer’s eruptions. After 15 weeks, though, another photo shows the man’s chest almost completely smooth, its healthy color restored. Lazarus, touched by vemurafenib. Then, two months later, the man’s chest is wan and bandaged, with protrusions so substantial that they are visible in profile at his neck. He died in hospice.

In June 2012, a year after Grace’s miraculous recovery, Maddox, the first of her three grandchildren, was born. The happy beginning of the next generation was all the more meaningful for Grace because she had not expected to witness it. But cancer doesn’t rest, and Grace’s doctors knew she faced the same threat as the man with metastatic melanoma. If a tumor cannot be eliminated, drug resistance is the rule. As the biological machinery goes awry inside a cancerous mass, reproducing cells make more errors copying their DNA, releasing a stream of mutants. The everolimus had made conditions harsh, but if a mutant should arise that could thrive despite the drug, it would divide, and divide again, eventually taking over and spreading. Darwin’s survival of the fittest governs inside a tumor, selecting for a crueler version of the disease. The cancer was looking for a new way out.

Grace is a pioneer, which means that others might learn from her. But it also means that her doctor, Jochen Lorch, has no precedents to guide him, because her case is without precedent. Within months of Maddox’s birth, he had to tell Grace that a substantial mass had appeared near her right lung and that several smaller nodules had taken root nearby. The tumor was “progressing” — surely one of medicine’s most perverse coinages.

When Grace became resistant to the everolimus, Lorch decided to try a pharmaceutical cousin, temsirolimus. (Resistant tumors sometimes stick to the same basic escape plan.) Lorch also asked if she would be willing to undergo a biopsy, so that the resistant tumor could be sequenced and compared with the original. This, he explained, might help future patients who faced everolimus resistance.

To understand that cancer is genetic is also to realize that the disease lurks within a biological system that has about 20,000 genes, each vulnerable in many ways. Some day, researchers hope to be able to develop a comprehensive resistance map for cancer, a full accounting of the ways that the disease, in all its varieties, defeats different therapies. The sequence of Grace’s resistant tumor could fill in one tiny part of such a map. Eventually, some scientists believe, treatments for most cancers could come in the form of multiple advanced therapeutics, applied in parallel. One would be aimed at containing the tumor’s current mutations, while the others would target its favorite backups. Hemmed in and blocked from shifting to Plans B or C or D, the insurrection would collapse.

When Lorch suggested the biopsy, Grace was preparing for her grandson Maddox’s dedication ceremony, and she knew that any surgical complications might put her attendance at risk. But her faith assured her that there was a reason she had happened into Lorch’s care — that any adversities she faced would eventually lead to a greater good. “She is deeply convinced that what we are doing here, and what is happening with her, is all part of a plan,” Lorch says.

At Maddox’s ceremony in November 2012, Grace’s ribs flared with every breath. The biopsy had gone smoothly, but the area was still tender. Grace used to be a standout in the choir, and one of her favorite hymns had always been “It Is Well With My Soul,” but since the radiation damaged her vocal cords, she had been unable to sing. A few months passed, and the temsirolimus wasn’t working. Lorch tried another drug; it didn’t work either.

One of the more grotesque insights yielded by cancer genetics is that many of the genes implicated in the disease are the same genes that guide early human development. All of us begin as a relatively formless embryo, and from there the cells follow an elaborate program. They grow rapidly. They migrate and specialize, taking on roles like nerve or skin or the production of hormones in the thyroid. Cells that are no longer needed die willingly. All of this activity is ordained in our DNA and orchestrated by an elaborate system of cellular communication in which genes activate and deactivate one another as needed. It is hard to imagine, but this cooperative dance is what allows an unshaped mass of cells to become something as perfect and graceful as a baby’s tiny hand, with each fingernail sculpted just so.

In Grace’s body, cancer hijacked this system and turned it to its own ends. For example, the gene that mutated in Grace’s original cancer, TSC2, is part of what’s called the mTOR pathway, which helps direct cellular growth. Normally the TSC2 gene acts as a brake, sending cease-and-desist orders to the mTOR gene when it’s time for the cellular engines to ease. The mutation trashed the brakes, locking the cells in overdrive. Everolimus, however, acts on the mTOR gene directly, limiting its ability to emit growth signals — an emergency brake. When Grace took it, her cancer faltered.

Early in 2013, Grace’s team received the sequence of her resistant tumor, and its trick was revealed: A single letter of DNA had changed in the mTOR gene. This solitary substitution meant that a lone amino acid, part of a long chain that makes up the mTOR protein, was different. Because of this, the mTOR protein assumed a slightly deformed shape, and this meant that the everolimus could no longer grab hold and do its work. The cancer roared to life. It sounds like truly extraordinary bad luck — a one-in-three-billion shot — but the power of evolution means that it’s entirely predictable: Inside the tumor, cancer rolls a pair of dice, over and over, until they come up snake eyes.

Often cancer appears to reverse the natural course of things, by taking mature cells, disciplined in form and function, and returning them to a more fevered, inchoate state. Anaplastic cancers like Grace’s are those in which the cells have lost their characteristic form. Seen on a pathology slide, normal thyroid follicular cells have a smooth, rounded look, like uncracked eggs, and they are arranged in neat circles. With anaplastic thyroid cancer, the cells swell up into a mess of irregular shapes, as if they’d all been melted together. They are reproducing so often that it’s not unusual for a pathology slide to show a cell in the middle of doubling, caught in the act.

Cancer is a monster, but in its fierce evolutionary tendencies, it is grasping, as with anything else in nature, for a way to be in the world. Life on earth has invaded the air, the deep sea, the bedrock. Over eons, it has suffered meteor storms, volcanic dystopias, shifting continents and deprivations beyond counting, and yet it always comes back stronger. With cancer, biology’s fierce insistence — its resilience, its ceaseless creativity, its sheer generative capacity — is the enemy. With cancer, the opponent is life.

As Lorch considered other ways to help Grace, he knew that one option might come from the pharmaceutical company Millennium, in Cambridge, Mass., which was working on a next-generation mTOR inhibitor, a drug that, like everolimus, targets the mTOR gene. By the summer of 2013, Wagle’s colleagues at the Broad Institute of M.I.T. and Harvard, where he has a position, had made some progress. After creating a proxy cancer, a collection of cells with the specific mutation found in Grace’s resistant tumor, they applied a compound that mimicked the activity of the drug by Millennium, which is now called Takeda Oncology. The drug should work, they concluded. The company told Lorch that it would support a clinical trial as soon as the drug was ready. Grace had agreed to the biopsy to help future patients, and now she had a chance to become one of them.

For Lorch, the challenge was to keep Grace alive long enough to join the trial. He knew her tumor was always in motion. Under the long press of everolimus, it had mutated, allowing it to resurge. The tumor then shook off two other drugs, including temsirolimus. Lorch decided to go with a traditional chemotherapy combination: carboplatin plus paclitaxel. In July 2013, he started Grace on the new therapy. As fall arrived, hair was slipping from her scalp, but the cancer had gone quiet, pharmaceutically stunned.

For all the excitement surrounding precision medicine, it is humbling to see how distant a goal it remains, even in cancer. There was nothing especially precise about the chemotherapy Lorch prescribed for Grace, and this, for most patients, is the reality today. Despite all of oncology’s recent successes, our understanding of the human cell’s vast genetic machinery — cancer’s playground — remains modest. The original Illumina sequence of Sharon’s tumor revealed a total of 17,136 mutations. Pick any one and inquire about its significance, and the answer will most likely be: Who knows?

To make the shift to precision oncology, cancer researchers have invented a novel means to evaluate new treatments. Called a basket trial, it is a trial for which patients are recruited by the tumor’s genetic signature, not its point of origin in the body; a drug is thus tested against a basket of many cancer types. Last August, an international team led by researchers at Sloan Kettering published some of the first results of such a trial, for cancers with a mutation in a gene called BRAF. The doctors saw a good response in lung cancer and two rare cancers, showing the power of selecting drugs based on tumor genetics. For other cancer types in the trial, though, the drug was less effective, or there were too few patients to draw conclusions, showing how much more remains to be learned. Sometimes targeting a mutation fails because it is only one of many driving the tumor, or because the mutation occurred at random in the genetic chaos, a passenger, while other mutations do the driving. Future basket trials could help doctors tease out the distinctions.

Many such trials are now underway, including a large federally supported effort called NCI-Match. Sharon and Grace’s cases even helped inspire an everolimus basket trial. Carole Arenson, an 80-year-old Illinois woman, has metastatic sarcoma with a mutation in TSC2, the same troublesome gene implicated in Grace’s disease. Carole joined the everolimus basket trial and, in November, she learned that the tumors were shrinking. In this, the scientific investigations can be seen coming full circle, from a pair of Lazarus miracles to practical medicine that, to Carole, has felt like salvation.

When I met Lorch in his office last year, he was optimistic. Grace had just started on the Takeda Oncology drug, and the initial signs pointed to considerable shrinkage. Back in the fall of 2010, when she first received her diagnosis, her life expectancy could be measured in months. “We are trying to turn [cancer] into a chronic disease,” Lorch said. “Right now, to have someone still alive five years after they were diagnosed and enjoying her grandkids is the best that we can do.” As we spoke, though, on his computer screen was a scan with spots: masses of Grace’s tumor cells, within which, concealed from him, the cancer was surely plotting its next escape. Lorch sat a bit sideways on his chair, leaning back with his hand a loose fist on his forehead, as if he were bracing for impact. “When her time comes, it is going to be hard to speak with her husband and all the people who’ve come in with her over the years.” He paused. “Am I dreading this? Yes. Does it motivate me to try harder to keep her around? Yes, absolutely.”

President Obama’s $202 million Precision Medicine Initiative, announced during his 2015 State of the Union address, seeks to study one million American volunteers to learn how genetic and other data might be used to tailor treatments. The initiative aims for progress in many ailments, including heart disease, diabetes, obesity and depression, but initially the focus is cancer. In January, in his final State of the Union address, Obama announced that he was putting Vice President Joseph R. Biden Jr. in charge of a “moonshot” to cure cancer. Beau Biden, the vice president’s son, died in May 2015, at age 46, of brain cancer. The president’s mother died of uterine and ovarian cancer. “For the loved ones we’ve all lost,” Obama said before the joint session of Congress, “for the families that we can still save, let’s make America the country that cures cancer once and for all.”

Two days after the president’s moonshot speech, I was on the eighth floor of the Charles A. Dana Building in Boston to visit Wagle at Dana-Farber. Last October, Wagle unveiled the Metastatic Breast Cancer Project, inspired partly by his involvement in Grace’s case and partly by frustration over the way cancer research works today. It’s quite difficult to track down patients with intriguing case histories, scattered as they are across the country and protected by blankets of privacy. So instead of going through doctors or hospitals, the project makes its appeal to patients directly. Through the project’s website, they can enter their medical histories and grant Wagle’s team access to their records, their DNA and tumor samples. Participants have started recruiting others to the project, solving a central challenge facing the scientific study of any rare phenomenon. In six months, more than 1,800 patients with metastatic breast cancer have joined, including hundreds of exceptional responders. In return, the project involves them in its decision-making and promises to share its data with any scientist who asks. “A lot of patients feel like the research-industrial complex is about making discoveries and competing for grants, and in many ways they are right,” Wagle said. “I get it.”

The project cuts against the grain of a medical system that was not designed to learn from patients. Every day in this country, doctors treat people for all kinds of disorders, and some do surprisingly well, or surprisingly poorly — and virtually all of this information is lost to science. Eric Lander, the founding director of the Broad Institute and co-chairman of the President’s Council of Advisors on Science and Technology, has begun laying the groundwork for a national project he calls Count Me In, which would allow anyone to make their medical records, and DNA, available to researchers. “In my opinion,” Lander said, “it’s a crime to let valuable information go to waste when a patient wants to share it.”

Wagle’s project is Count Me In’s first undertaking. Next, Count Me In is beginning a similar effort on angiosarcoma, an understudied cancer, and two more efforts are planned this year. Precision oncology has been driven by advances in two areas: automation, responsible for the plummeting cost of genetic sequencing, and information technology, which allows the data to be recorded and interpreted. Projects like Count Me In are built on the premise that a third disruptive technology can also be brought to bear: social media.

The campaign by American political leaders to cure cancer goes back decades, including Nixon’s “war on cancer.” But these latest federal calls for action — which bring to mind images of the best and brightest, backed by brute force — risk mistaking the nature of the opponent and misunderstanding the potential of the weapons now available. Ask almost any specialist about a cure for cancer, and she will cringe, because there is no one cancer to be cured. We face an enemy that is resourceful, changeable and merciless, but we have a population that wants to help. Instead of a war (or a moonshot), what is required probably looks more like a counterinsurgency operation. The people are tired of being victimized, unable to have a say in their own fates. Almost any one of them would tell you: If you are going to die, it’s better to die with a purpose.

The patients enlisted so far, however, have joined almost by accident. Sharon, who continues to live in Florida without any sign of cancer, explained how thoroughly her good fortune has depended on her circumstances. Her son-in-law, an oncologist, pointed her to Sloan Kettering, and she could afford to fly up every four weeks to participate in the trial. “I think about the people who would have trouble with funds, and it breaks my heart,” she said. If Grace had not happened to live near Boston, the odds that she could have laid eyes on any of her grandchildren — or contributed to science — are long indeed. Carole Arenson, who has continued to do well on everolimus, found a place in the basket trial only because she had Foundation Medicine, a company in Cambridge, Mass., sequence her tumor, revealing the crucial mutation. “I’m as happy to see you,” Carole recalled telling a doctor involved with the trial, “as you are to see me.”

This haphazard approach has led to a chicken-and-egg problem that is among the most obvious obstacles to progress in precision oncology. In order to run basket trials, researchers must find patients with the right mutations. But this has proved difficult, because most patients aren’t having sequences done: Insurance generally doesn’t cover it. The reason? Because, insurance companies point out, there is not yet enough evidence that it will be clinically useful. The way to gather this evidence? Basket trials.

José Baselga, the physician in chief and chief medical officer at Sloan Kettering, told me that his hospital has a number of quite promising basket trials running but is struggling to find enough patients. One analysis published last year found that patients in genetically targeted trials are seeing better outcomes than those in traditional trials. Yet if patients who would qualify aren’t fortunate enough to be at cancer centers like Sloan Kettering or Dana-Farber that run sequencing programs supported by philanthropic money, they either need to pay for it themselves or they are out of luck — and so is cancer research. Baselga, in conversations with Biden, has argued for a change in policy: If Medicare covered sequencing, private insurance would follow, opening up precision medicine to many more people.

It’s a problem of scale. Sloan Kettering, Dana-Farber and several other top cancer hospitals are pooling their findings so that oncologists can search through cases, about 20,000 so far, to see what mutations were found, what drugs were tried and how patients responded. The National Cancer Institute’s Exceptional Responders Initiative has accumulated about 50 confirmed cases from around the country. Fifty exceptional responders is a large number, but it is also small. The same could be said of 20,000 cancer cases, annotated with genetic and clinical information: large, unprecedented and also not nearly enough.

For thousands of years, deep thinkers about military strategy have understood that wars are not won in the way the public imagines. Gen. Robert H. Barrow, who served four years as commandant of the United States Marine Corps, is credited with saying that “amateurs talk about tactics, but professionals study logistics.” Science has revealed the nature of cancer, and also created new means to gather intelligence on the foe. Solitary engagements can reveal new weaknesses to exploit. Does anyone truly imagine prevailing without bringing the fight everywhere — without matching cancer in its inventiveness, its nimbleness, its sheer relentlessness?

On a bitter morning in January, I visited Grace at home. When I stepped into the entryway, she stood at the top of some steps, wearing a purple top, black leggings and a pair of comfy black slippers. From a television, I made out a voice that could be only Elmo. Grace smiled.

She took a seat on a sofa with Jaelynn, her 2-year-old granddaughter, who sat in an off-white cable-knit sweater watching the TV and occasionally pushing her fine black hair from her face. On the wall hung a photograph of Faial, the Portuguese island where Grace’s father, who went by Mestre Simao, had been a ferry captain. Simao was beloved for braving severe storms to fetch people in distress from a nearby island and deliver them to the Faial hospital. He would tie himself to the wheel. Sometimes he returned with a cockpit window blown out.

Grace showed me a video on her phone from 2013, when a new ferry arrived at the island: The Mestre Simao, named for her father. With its lime green and dark blue bow, the boat made its way in from the ocean, trailed by a tug. Jaelynn stood next to Grace, enraptured as the scene unfolded, a hand resting on her grandmother’s leg.

Sitting there, I found myself wondering if something happened during one of Simao’s storm runs. I imagined a gravely ill man, tied down for safety, the sea enraged. The man would have been praying, and he would have known that his only earthly hope was Simao, up in the cockpit, refusing to panic as the marine glass shattered in his face. Was there a moment when this was somehow noted — when some beneficence was granted to Simao’s youngest, little Grace, who always took his lunch to the docks?

The trailing tug began pumping ocean water, two celebratory arches in the air. “Water!” Jaelynn shouted. “I know, I know,” Grace said. “Look, it’s so pretty.” The boats circled. “You want to go someday on a ship?” Grace asked. Jaelynn’s eyes kept to the screen. It was morning in the Azores, and the light hit low. “Yeah,” she said.

A few months later, Grace began coughing up blood. Joe rushed her to the emergency room: A tumor in her lung was growing again. The Takeda Oncology drug had been good for six months. Grace went through a round of radiation on the lung and then started nivolumab, a therapy that helps the immune system attack tumors. Earlier this month, though, she went to the hospital with dizziness and a severe headache. Scans showed metastasis to the brain.

Lying flat in a bed on the 12th floor of Brigham and Women’s Hospital, Grace assured Lorch that her minister was praying for him too, as he pondered his options: for her but also for what science might still learn from her remarkable case. The nivolumab was out, but Lorch did notice that the drug seemed to have been working around her lung — several tumors shrank — which is significant because doctors don’t know who will respond to the drug. He and his colleagues plan to take what they’ve learned about the genetics of Grace’s cancer and compare it with evidence, from her blood, of the drug’s activity. This might help the next patient. Grace may have yet more to give.

Gareth Cook is a contributing writer for the magazine and a Pulitzer Prize-winning journalist. He last wrote about the mathematician Terry Tao.