Milwaukee police officers were called to the Ambassador Hotel on May 6, 1993 to remove James Phillips, III from his room. When the officers arrived, he put up a struggle that escalated until the officers handcuffed Mr. Phillips and placed him face down on the floor. Mr. Phillips then stopped breathing and died the next day. His mother, Raye Phillips, as special administratrix of her son’s estate and on her own behalf, brought this suit alleging that the officers used excessive force in seizing her son and that the City of Milwaukee was also liable for having an unconstitutional policy allowing such force and for fading to train its officers properly. The district court granted summary judgment to the defendants. For the reasons given in the following opinion, we affirm.
I
BACKGROUND
A. Facts
On May 5, 1993, Mr. Phillips was evicted from the Ambassador Hotel by the general *588manager, Rick Scott. Prior to his eviction, Mr. Phillips had been behaving in a strange and disorderly manner. For example, he had placed furniture in the hallway in front of his room and had thrown a chair out of his window. He also had stuffed his clothes into the bathtub while running the water, thereby overflowing the tub. It was because of Mr. Phillips’ disruptive behavior that Scott asked Mr. Phillips to leave the hotel.
Mr. Phillips initially complied with Scott’s request, but he returned the next morning, May 6. With a key he apparently had retained, Mr. Phillips reentered his old room undetected. The head of housekeeping was on duty that morning and went into Mr. Phillips’ old room to check the condition of the room’s carpeting. When the employee entered the room, she encountered Mr. Phillips, who was naked at the time. She became frightened and ran out of the room. In response, Mr. Phillips attempted to barricade himself in the room by locking and chaining the door. Gary Kassin, a security guard at the hotel, arrived on the scene and tried to coax Mr. Phillips into unlocking the door. Mr. Phillips was uncooperative, and Kassin eventually unlocked the door with a master key and then had to cut the chain on the door.
When Kassin and James Rice, another hotel employee, entered the room, Mr. Phillips was wearing a trench coat, but was otherwise naked. Kassin testified that Mr. Phillips “was just going nuts” and was shaking “like a leaf.” Kassin Dep. at 15. Kassin noticed that Mr. Phillips had thrown the fans that were drying the carpet out of the window and had damaged other property in the room. Drawers were broken; furniture was moved to barricade the door; the sheets were torn off the bed and were soaking wet; the bathroom was in shambles; in short, “the place looked like a disaster area.” Id. at 16. Kassin asked Mr. Phillips to put on his pants and called down to the desk to have hotel personnel call the police. Kassin and Rice then sat with Mr. Phillips while they waited for the police to arrive.
The first Milwaukee police officers on the scene were defendants Dennis Hintz and Mary Riley. The desk clerk at the hotel- told the officers about the troubles the hotel had experienced with Mr. Phillips. The officers were informed that Mr. Phillips had been behaving strangely the past two days, that he had been evicted for flooding his room, and that he had just thrown fans out of his window to the parking lot below. The officers proceeded to the third floor where they encountered a security guard who reiterated that Mr. Phillips was behaving in a bizarre fashion. The officers then entered the room. Mr. Phillips was wearing a trench coat, no shirt, and trousers that were about to fall down; he appeared unkempt and soiled and was visibly shaking and sweating. Officer Hintz walked to the window and observed two fans lying in the parking lot. Officer Riley noticed that a framed picture, too large to throw out of the window, had been moved as if Mr. Phillips had attempted to throw it out as well.
Officer Hintz asked Mr. Phillips what his name was and what was going on, but got no response. The officer told him to pull up his pants, which Mr. Phillips did. Officer Riley then asked several questions of her own, but Mr. Phillips responded only with unintelligible sounds. At that point, one of the hotel employees told the officers to be carefid because Mr. Phillips had a ballpoint pen clenched in each hand. Kassin testified that, given Mr. Phillips’ size and the manner in which he held the pens, Mr. Phillips could have caused injury to the officers if he had lunged at them with the pens. Officer Hintz feared for his safety and grabbed one of Mr. Phillips’ wrists; Officer Riley grabbed the other. Mr. Phillips resisted, and a struggle ensued.
The struggle, though intense, was not marked by any violent behavior on the part of the police. Mr. Phillips was strong and refused to drop the pens; he struggled to break free of the officers’ grasps. For then-part, the officers could not pry Mr. Phillips’ fingers apart, so the pens remained in Mr. Phillips’ clenched fists. As a result, Officer Hintz decided that Mr. Phillips should be placed on the bed to control him without injuring anyone. According to Kassin, the officers were not shouting or being abusive. They were telling Mr. Phillips that every*589thing would be fíne and that they were just going to take him out of the hotel. It appeared to Kassin that the officers were trying to help Mr. Phillips. Mr. Phillips, on the other hand, became extremely violent when the officers tried to move him over to the bed. He began thrashing his arms and pulling with his shoulders and upper body. As a result, the officers decided to place Mr. Phillips on the floor instead. With the aid of hotel personnel, the officers ■ succeeded in bringing the struggling Mr. Phillips to the floor. Once there, Mr. Phillips became more violent, lacked his feet and tried to bite Officer Riley. The hotel staff held Mr. Phillips’ legs while each officer managed to place a pair of handcuffs on each of Mr. Phillips’ wrists. Due to the size of Mr. Phillips, the officers connected the handcuffs so that his arms were restrained behind his body by two sets of handcuffs connected in the middle.
Mr. Phillips continued to struggle and kick violently while on the floor. Officer Riley placed her knee on Mr. Phillips’ right shoulder blade to hold him down and to keep him from turning over and kicking. Because of the intensity of Mr. Phillips’ struggling, Officer Hintz was certain that Mr. Phillips could have turned himself over absent the pressure Officer Riley was applying. Kassin testified that Officer Riley’s knee was “gently sitting on [Mr. Phillips] so he couldn’t raise up” for about thirty seconds. Kassin Dep. at 28, 55. Officer Riley estimated that she had her knee on Mr. Phillips’ back for about a minute. Kassin recalled further that Officer Riley did not have much of her weight on Mr. Phillips as she almost lost her balance.
Based on his observations, Officer Hintz was of the opinion that Mr. Phillips needed mental observation at a hospital, not incarceration. He called for a patrol wagon to transport Mr. Phillips to the Milwaukee County Medical Complex. Officer Riley then observed Mr. Phillips reaching in his pockets. Two dinner forks were found in Mr. Phillips’ coat pocket and were removed from his possession. As a result of Mr. Phillips’ continuing struggle, Officer Hintz decided to call an ambulance. His thought was that it would be safer to transport Mr. Phillips on a stretcher to keep Mr. Phillips from injuring himself.
At that time, Officer Duarte arrived at the scene as backup. Officer Riley testified that Mr. Phillips was in a face down, handcuffed position for about one minute prior to Officer Duarte’s arrival. Officer Duarte held Mr. Phillips’ feet to prevent them from kicking. He was asked to call an ambulance by Officer Hintz and did so by radio. A few minutes after the arrival of Officer Duarte, Officer Busch arrived with leg restraints and placed them on Mr. Phillips’ ankles, at which time Mr. Phillips’ legs were still moving. Kassin estimated that the leg restraints were applied about a minute or so after the handcuffs. Mr. Phillips was not “hog-tied”; that is, the leg restraints and handcuffs were not connected together. After a few minutes, Mr. Phillips began to calm down, and the officers heard the ambulance crew in the hall. Thinking that Mr. Phillips would remain calm if he knew that he was not going to jail, Officer Hintz told Mr. Phillips that an ambulance had arrived to take him to the hospital. Kassin testified that, during the time Mr. Phillips was on the floor, the officers constantly watched him and called his name every twenty or thirty seconds; although on this latter point, Officer Riley could not recall anything being said to Mr. Phillips until Officer Hintz informed Mr. Phillips that the ambulance had arrived.
The record is unclear as to who determined that Mr. Phillips had stopped breathing. Kassin and several of the officers say that Officer Duarte made that determination after rolling Mr. Phillips over and checking his pulse. The ambulance attendants say that they initially made the discovery; that is also Officer Busch’s recollection. The ambulance attendants, Edward Meade and Jay Sehiellack, arrived within four minutes of receiving the dispatch to transport Mr. Phillips for mental observation. Officer Busch says that they arrived about thirty seconds to one minute after he had applied the leg restraints. Meade testified that, when he entered the room, he observed that the officers were standing within about a foot of Mr. Phillips. Meade asked the officers some questions about the situation and then, about *590a minute and a half after entering the room, knelt to ask Mr. Phillips a question. Mr. Phillips did not respond; at that time Meade determined that Mi*. Phillips was a “pulseless nonbreather.” Meade testified that he then left the room to get additional equipment and to call the paramedics; having not anticipated Mr. Phillips’ situation, he had left the necessary equipment in the ambulance. Officer Busch testified that he cleared the room of all non-police personnel for a minute or two when Meade announced Mr. Phillips’ status because, at that point, the scene was a possible homicide scene. Officer Busch says that only the ambulance personnel and paramedics were allowed back in the room. Meade says simply that he left voluntarily to get more equipment and returned to the room without any problem. In any event, during the time that Meade was gone, the officers performed CPR in an attempt to revive Mr. Phillips. When the ambulance attendants returned, they continued administering CPR. After a few minutes, one of the ambulance crew became tired, and an officer took over chest compressions for him. The paramedics arrived soon thereafter and took over the care of Mr. Phillips. Mr. Phillips regained a pulse.
Mr. Phillips was taken to Sinai Samaritan Hospital. He died the next day. Dr. Steven Kream diagnosed Mr. Phillips’ condition as involving cardiac arrest, anoxic encephalopa7 thy/probable brain death, thyroid storm, Graves’ disease and schizophrenia. Dr. Kream explained that these were contributing factors leading to Mr. Phillips’ death. Dr. Kream opined that brain death can occur within three to four minutes of low oxygen to the brain and that brain death can occur even sooner in a person like Mr. Phillips who has suffered from disease. A thyroid storm results in extreme overactivity; a person with that condition may experience sweating, rapid breathing, a rapid pulse, and an active nervous system and heart. According to Dr. Kream, the manifestations of a thyroid storm are not necessarily observable. A thyroid storm left untreated can cause death.
Dr. Jeffrey Jentzen performed an autopsy on Mr. Phillips on May 8. He determined that Mr. Phillips was an extremely obese individual. In his deposition, Dr. Jentzen explained that, when an obese person is placed on the floor in a prone position, his ability to breathe is inhibited. Dr. Jentzen also took note of Mr. Phillips’ history of schizophrenia and Graves’ disease. He found that Mr. Phillips’ heart was moderately enlarged and that Mr. Phillips’ thyroid gland was extremely enlarged and almost wrapped completely around Mr. Phillips’ trachea. Dr. Jentzen opined that a reasonable individual or police officer would not have been able to detect Mr. Phillips’ heart and thyroid problems upon observation. The autopsy revealed some minor abrasions on Mr. Phillips’ body, but there were no external injuries to the head, scalp or neck region. Nor did the doctor observe any hemorrhaging in body cavities to suggest trauma. The death certificate says that the cause of death was “[s]ud-den unexpected death associated with law enforcement restraint.” R.29, Ex.28. Dr. Jentzen explained that this “cause of death” reflected the totality of the circumstances surrounding Mr. Phillips’ demise. He explained that placing'Mr. Phillips in a prone position with his hands behind his back may have been a contributing factor to Mr. Phillips’ death. Mr. Phillips’ obesity and other physical problems also contributed. Dr. Jentzen additionally noted that Mr. Phillips had not been taking his medication for his psychiatric condition.
The officers testified that leaving Mr. Phillips face down while handcuffed comported with the Milwaukee Police Department’s policies. Since Mr. Phillips’ death, the policy has been amended; officers are now told to seat or stand an individual as soon as possible once the individual is restrained.
B. District Court Proceedings
The district court granted summary judgment to the defendants, primarily on the ground of qualified immunity. See Estate of Phillips v. City of Milwaukee, 928 F.Supp. 817 (E.D.Wis.1996). Noting that all claims of excessive force by law enforcement officials are analyzed under the Fourth Amendment, see Graham v. Connor, 490 U.S. 386, 395, 109 S.Ct. 1865, 1871, 104 L.Ed.2d 443 (1989), the court further concluded that the seizure in *591this case was reasonable. The court noted that the officers properly had grabbed Mr. Phillips’ hands to release the pens and that the struggle then had escalated. The take-down was reasonable, as was the use of leg restraints and handcuffs, the court stated, because of Mr. Phillips’ aggressive behavior. The court further determined that, after Mr. Phillips had been cuffed, he had been monitored by the officers. According to the district court, the record demonstrated that only one or two minutes passed from the time Mr. Phillips stopped struggling to the time the ambulance arrived. The events that followed — Meade’s investigation, acquiring medical equipment, beginning CPR — transpired in a matter of minutes. Because no question of material fact existed as to whether the force used was excessive or whether the officers knew Mr. Phillips would stop breathing, the court decided there was no possibility that the officers acted unreasonably under the circumstances.
Upon reaching the plaintiffs’ claims against the City of Milwaukee and Police Chief Philip Arreola, the court opined that its finding that the officers did not violate Mr. Phillips’ constitutional rights mooted the question whether the policies of the City were unconstitutional. In any event, the court held, the plaintiffs could not prevail on these claims because they could not show that the failure to train amounted to deliberate indifference. The court determined that the plaintiffs’ submissions — a letter, articles and reports discussing in-custody deaths — allegedly placing the City on notice involved situations different from the one in which Mr. Phillips was found. The arrestees in those cases, for example, had been hog-tied, transported in a prone position, confined in a small space or placed in a choke hold. The court also took the view that none of the prior in-custody deaths that had occurred in the City was similar to Mr. Phillips’ death: two from cocaine, one from natural causes, and two from suicide.
The district court retained jurisdiction over the plaintiffs’ state law claims in the interest of judicial economy because, in its view, Wisconsin law clearly foreclosed their claims. Police officers are generally immune under Wisconsin law from liability for injuries resulting from acts performed within the scope of the officers’ duties. Barillari v. City of Milwaukee, 194 Wis.2d 247, 533 N.W.2d 759, 763 (1995). The court concluded that none of the three exceptions to the general rule of immunity applied here: The officers had not (1) engaged in malicious, willful or intentional conduct, (2) negligently performed a ministerial duty, or (3) had an absolute duty to act because they were aware of a danger justifying such a duty. Because the court had granted summary judgment to the police officers and the City on the plaintiffs’ federal claims, it dismissed the claims against the remaining defendants, James Rice and Harold Ray Schmidt, as well.
II
DISCUSSION
Summary judgment is appropriate if the materials in the record “show that there is no genuine issue as to any material fact and that the moving party is entitled to a judgment as a matter of law.” Fed. R. Civ. P. 56(c). A party is entitled to such a judgment if, based on the evidence in the record, no reasonable jury could return a verdict for the non-moving party. Anderson v. Liberty Lobby, Inc., 477 U.S. 242, 248-52, 255, 106 S.Ct. 2505, 2509-10, 2513, 91 L.Ed.2d 202 (1986). “A party must present more than mere speculation or conjecture to defeat a summary judgment motion.” Sybron Transition Corp. v. Security Ins. Co., 107 F.3d 1250, 1255 (7th Cir.1997). We review the district court’s decision to grant summary judgment de novo. Id.
A.
At the outset, we note what is not at issue in this case. The plaintiffs do not claim that the police officers lacked the requisite justification for seizing Mr. Phillips; they concede that the officers were within their rights to take Mr. Phillips into custody. Nor do they contend that the officers used excessive force in grabbing Mr. Phillips’ wrists and taking Mr. Phillips to the floor. Their claim is that the excessive foree occurred during the few minutes that Mr. Phillips was on the floor in *592a prone position. In the plaintiffs’ view, the officers placed Mr. Phillips in a precarious position, thereby depriving him of sufficient oxygen, without sufficiently monitoring him to determine if he was breathing.
Excessive force claims arise under the Fourth Amendment. Tennessee v. Garner, 471 U.S. 1, 8, 105 S.Ct. 1694, 1699, 85 L.Ed.2d 1 (1985) (“[I]t is plain that [the] reasonableness [of a seizure under the Fourth Amendment] depends on not only when a seizure is made, but also how it is carried out.”). In Graham v. Connor, 490 U.S. 386, 109 S.Ct. 1865, 104 L.Ed.2d 443 (1989), the Supreme Court held that “all claims that law enforcement officers have used excessive force — deadly or not — in the course of an arrest, investigatory stop, or other ‘seizure’ of a free citizen should be analyzed under the Fourth Amendment and its ‘reasonableness’ standard.” Id. at 395, 109 S.Ct. at 1871 (emphasis in original). See generally Lanigan v. Village of East Hazel Crest, 110 F.3d 467, 474-76 (7th Cir.1997) (discussing Graham); Frazell v. Flanigan, 102 F.3d 877, 882-83 (7th Cir.1996) (same); Brownell v. Figel, 950 F.2d 1285, 1292-93 (7th Cir.1991) (same). The reasonableness standard cannot be defined precisely, nor should it be applied in a mechanical fashion. Graham, 490 U.S. at 396, 109 S.Ct. at 1871. “Determining whether the force used to effect a particular seizure is reasonable under the Fourth Amendment requires a careful balancing of the nature and quality of the intrusion on the individual’s Fourth Amendment interests against the countervailing governmental interests at stake.” Id. (internal quotations and citation omitted). Due to the fact-specific nature of the inquiry, the determination whether a police officer utilized excessive force depends on the totality of the circumstances surrounding the encounter. Frazell, 102 F.3d at 882. We must pay “careful attention to the facts and circumstances of each particular case, including the severity of the crime at issue, whether the suspect pose[d] an immediate threat to the safety of the officers or others, and whether he [was] actively resisting arrest or attempting to evade arrest by flight.” Graham, 490 U.S. at 396, 109 S.Ct. at 1871 (citing Garner, 471 U.S. at 8-9, 105 S.Ct. at 1699-700). In addition, the reasonableness inquiry is an objective one: The “particular use of force must be judged from the perspective of a reasonable officer on the scene, rather than with the 20/20 vision of hindsight.” Id. “[T]he question is whether the officers’ actions are ‘objectively reasonable’ in light of the facts and circumstances confronting them, without regard to their underlying intent or motivation.” Id. at 397, 109 S.Ct. at 1872.
Notwithstanding that the plaintiffs concede that the force used was reasonable up until the point that Mr. Phillips was placed in a prone position on the floor, we begin our analysis with the moment the officers arrived on the scene in order to place the officers’ conduct in context. When Officers Hintz and Riley initially arrived at the hotel, they were informed that a man was exhibiting erratic and disorderly behavior, damaging property and throwing items out of a window to the ground three floors below. Their own initial observation of Mr. Phillips confirmed what they had been told. Mr. Phillips was shaking like a leaf and sweating profusely. The officers noticed that the room was in disarray and that the fans had indeed been jettisoned from the window. Mr. Phillips was unresponsive and appeared agitated. Faced with this situation, the officers were entirely reasonable in grabbing Mr. Phillips’ wrists when informed that he had two ballpoint pens in his fists. Kassin testified that, due to Mr. Phillips’ size and the manner in which he held the pens, Mr. Phillips could have caused severe injury with the pens. The officers were reasonable in trying to protect themselves and the hotel employees by disarming Mr. Phillips.
From that point, the struggle escalated, but the uncontroverted testimony is that Mr. Phillips’ actions caused the escalation. He thrashed about violently when his wrists were grabbed and refused to unclench his fists to drop the pens. The officers attempted to soothe him with words and to take him over to the bed; indeed, Kassin testified that it appeared that the officers were merely trying to help Mr. Phillips. Yet the officers had little success in this regard. Mr. Phillips *593was strong, and the officers could not remove the pens, move him to the bed or get his arms behind him. So the officers understandably decided to take Mr. Phillips to the floor, which required that his legs be taken out from under him. This maneuver was accomplished with the aid of hotel employees.
Once on the ground, Mr. Phillips continued to struggle. He tried to roll over and kick the officers, and he at one point attempted to bite Officer Riley. The officers’ response was reasonable. Officer Riley placed just enough weight on Mr. Phillips to keep him from rolling over and kicking. If she had not held Mr. Phillips down, the witnesses testified, Mr. Phillips could have gotten up again and would have been a danger to himself, the officers and the hotel employees. Almost immediately after Mr. Phillips was on the ground, Officer Duarte arrived as backup; he confirms that Mr. Phillips was still moving at that point. Officer Duarte then called an ambulance and took up a position at Mr. Phillips’ feet to protect those around Mr. Phillips from his kicking. Officer Busch arrived immediately thereafter and solved the kicking danger by placing restraints on Mr. Phillips’ legs. The restraints placed on Mr. Phillips’ legs were similar to handcuffs and allowed the legs to move apart to some degree; the leg restraints were not connected to his handcuffs so as to place Mr. Phillips in a hog-tied position. It was reasonable to restrain Mr. Phillips’ legs given the peril posed by his continued kicking. The testimony as to the timing of these events varies somewhat, but the difference is immaterial. It is undisputed that just a few minutes passed between Mr. Phillips’ wrists being cuffed and his legs being restrained, and at any rate, Mr. Phillips was still moving at the time the leg restraints were applied.
We hold that these actions of the officers amount to an objectively reasonable response to the escalating situation they faced. “Authorities must be allowed ‘to graduate their response to the demands of any particular situation.’ ” United States v. Montoya de Hernandez, 473 U.S. 531, 542, 105 S.Ct. 3304, 3310, 87 L.Ed.2d 381 (1985) (quoting United States v. Place, 462 U.S. 696, 709 n. 10, 103 S.Ct. 2637, 2646 n. 10, 77 L.Ed.2d 110 (1983)); see Lanigan v. Village of East Hazel Crest, 110 F.3d 467, 475-76 (7th Cir.1997) (indicating that the amount of force that is justified increases as the confrontation escalates). Mr. Phillips was placed in a prone position with his hands and legs restrained because of the need to incapacitate him and to protect the safety of the officers and other witnesses from the dangers posed by Mr. Phillips’ violent behavior. Restraining a person in a prone position is not, in and of itself, excessive force when the person restrained is resisting arrest. Mayard v. Hopwood, 105 F.3d 1226, 1227-28 (8th Cir.1997) (holding that officers’ actions were objectively reasonable as a matter of law when they handcuffed resisting individual’s wrists, placed hobble restraints on individual’s legs and placed individual face down in squad car). The medical evidence and witness testimony in this case shows that the officers did not punch, slap, kick or otherwise deliver a blow to Mr. Phillips’ body. Cf. Frazell v. Flanigan, 102 F.3d 877, 885 (7th Cir.1996) (“Yet it is one thing to use force in subduing a potentially dangerous or violent suspect, and quite another to proceed to gratuitously beat him.”). Kassin’s testimony confirms that the officers were not beating Mr. Phillips, but were assuring him and trying to be of assistance.
Our holding that it was reasonable to place Mr. Phillips on the floor in a prone position necessarily implies that the officers did not use “deadly” force to restrain Mr. Phillips. Given the situation here, the officers would not have been justified, and they concede as much, in using deadly force to arrest Mr. Phillips. See Tennessee v. Garner, 471 U.S. 1, 105 S.Ct. 1694, 85 L.Ed.2d 1 (1985) (holding that deadly force is appropriate only in certain situations). For a particular application of force to be classified as “deadly,” it must at least “carry with it a substantial risk of causing death or serious bodily harm.” Robinette v. Barnes, 854 F.2d 909, 912 (6th Cir.1988) (internal quotation omitted); see Donovan v. City of Milwaukee, 17 F.3d 944, 949-50 (7th Cir.1994). Under this standard, restraining a person in a prone position, with constant monitoring, cannot be characterized, *594in itself, as “deadly” force.1 Here, the officers did not hog-tie, choke or transport Mr. Phillips. Nor were his medical conditions (an enlarged heart, an enlarged thyroid, Graves’ disease and a thyroid storm), which were contributing factors to Mr. Phillips' death, observable to the untrained eye. See Frazell, 102 F.3d at 885 (noting that officer’s conduct in the face of knowledge of suspect’s medical problems may be objectively unreasonable, although same conduct might be reasonable absent such knowledge). None of the plaintiffs’ materials supports that restraining an individual in a prone position “earr[ies] with it a substantial risk of causing death or serious bodily harm.” Robinette, 854 F.2d at 912 (internal quotation omitted). The officers placed Mr. Phillips in a face down position to restrain him from injuring himself and others. That force, it turned out, when combined with Mr. Phillips’ other health problems, resulted in Mr. Phillips’ death. But the question is not whether the officers’ actions aggravated an undiscovered injury or condition, but whether their actions were objectively reasonable under the circumstances. See Brownell v. Figel, 950 F.2d 1285, 1293 (7th Cir.1991). Placing Mr. Phillips in a prone position was reasonable under the circumstances and therefore comported with the Fourth Amendment.
Having decided that placing a person in a prone position while handcuffed on the floor does not, in and of itself, violate the Fourth Amendment, we must address whether the officers’ conduct while Mr. Phillips was on the floor can be deemed unreasonable. The plaintiffs claim that it was not reasonable for the officers to place Mr. Phillips in a position that denied him oxygen and then fail to notice that he was having trouble breathing. The record, though, will not support an inference that the officers failed to take reasonable steps to monitor Mr. Phillips. Although the plaintiffs contend that Mr. Phillips stayed on the floor without moving for five minutes, the testimony to which the plaintiffs invite our attention shows that Mr. Phillips was still moving when Officer Busch applied the leg restraints and that the ambulance crew arrived a minute or so after-wards. In any event, everyone agrees that Mr. Phillips was on the floor for only a few minutes before the ambulance crew arrived. More importantly, the witnesses’ testimony is also clear that the officers stayed right by Mr. Phillips during those few minutes. Some testimony indicates that two officers knelt on each side of him, while the other officers stood right beside him. Kassin testified that everyone in the room kept an eye on Mr. Phillips to make sure that he was calming down and was not injuring himself or others: “[Wje’re all watching, I mean, constantly they’re watching him because he was struggling so hard we knew he was tired, so we thought he was resting.... [H]e was moving a little bit, but you could see him breathing.... [Everybody is watching him, you know, watching him laying on the floor.” Kassin Dep. at 30. Kassin also testified that the officers called Mr. Phillips’ name every twenty or thirty seconds to make sure that Mr. Phillips was responding. Even though he alone reports this piece of information, no one reports that the officers failed to monitor Mr. Phillips.
The plaintiffs concede as much, but they maintain that the monitoring must have been insufficient because of the tragic result of Mr. Phillips’ encounter with the officers. Despite any evidence in the record that the officers failed to monitor Mr. Phillips in a manner that was reasonable under the tense and rapidly moving circumstances they faced, the plaintiffs say that Mr. Phillips’ demise is all the evidence they need of unreasonableness. The plaintiffs’ claim, therefore, essentially rests on “logic akin to the doctrine of res ipsa loquitur.” Brownell v. Figel, 950 F.2d 1285, 1292 (7th Cir.1991). In Brownell the plaintiff “maintain[ed] that the fact he left the hospital able to move and returned a quadriplegic” raised an inference “that the *595officers at the lockup employed excessive force.” Id. at 1292. Yet we affirmed the district court’s decision to grant summary judgment to the defendants because, “in the absence of any evidence of specific wrongdoing, Brownell’s excessive force claim [was] speculative and fail[ed] to raise any material issue of fact for trial.” Id. at 1293. Here, the plaintiffs’ claim similarly is speculative. Absent any evidence to prove that the officers in this case were unreasonable in their handling of Mr. Phillips once he was cuffed on the floor, the reasonableness question ought not be submitted to a jury.
It is true that the evidence of who discovered Mr. Phillips’ difficulties is not one-sided, that there is an issue of fact on that point. Kassin, for example, testified that, when Mr. Phillips did not respond to his name being called, an officer determined that Mr. Phillips was a pulseless nonbreather and began performing CPR on him. On the other hand, the ambulance attendants testified that, when they arrived, the officers had not yet realized that Mr. Phillips had stopped breathing; and Officer Busch agreed that the attendants were the first to discover that Mr. Phillips was a pulseless nonbreather. But who discovered that Mr. Phillips was not breathing is a fact of no consequence. The evidence remains uneontroverted that the officers monitored Mr. Phillips while he was on the floor. Even the ambulance attendants, on whose testimony the plaintiffs rely, testified that the officers were standing within about a foot of Mr. Phillips when they arrived. The fact also remains that there is no testimony that the officers ignored Mr. Phillips while he was in a prone position or that they walked away from him while he was on the floor. That no one noticed that Mr. Phillips was not getting enough oxygen cannot, standing alone, transform these officers’ actions into a transgression of constitutional magnitude. Indeed, even the trained ambulance attendant had been in the room for a minute and a half before he determined that Mr. Phillips was not breathing. Under these circumstances, to find that the officers should have monitored Mr. Phillips more carefully would be to judge the officers’ actions with the improper benefit of 20/20 hindsight. See Graham, 490 U.S. at 396-97, 109 S.Ct. at 1871-72 (also stating that “[t]he calculus of reasonableness must embody allowance for the fact that police officers are often forced to make split-second judgments — in circumstances that are tense, uncertain, and rapidly evolving....”).
Finally, there is the issue of Officer Busch’s testimony that he cleared the room of all non-police personnel (which ostensibly would include the health care personnel) after it was discovered that Mr. Phillips was a pulseless nonbreather. This testimony does not create a triable issue on the reasonableness of the officers’ actions. First, at oral argument, counsel for the plaintiffs expressly stated that he was not claiming that the officers violated Mr. Phillips’ constitutional rights by cutting off aid to him. Second, during the two minutes that the room was cleared, the officers performed CPR on Mr. Phillips, thereby earing for his medical needs while the ambulance crew was absent. Third, the ambulance personnel say that they left the room voluntarily after determining that Mr. Phillips was not breathing to get more equipment from the ambulance; because Mr. Phillips’ situation had been unanticipated, they had left behind the necessary life-saving equipment. As soon as they returned, they were allowed back into the room without problem. In other words, even in the light most favorable to the plaintiffs, Officer Busch’s command for non-police personnel to vacate the room must have corresponded with the ambulance crew’s voluntary decision to leave the room for more equipment.
We have been assuming that the police officers had a duty to provide medical attention (and not to cut off medical aid) when they seized Mr. Phillips. Yet it is somewhat awkward to conceptualize such an act or failure to act as “excessive force”; indeed, the duty to render medical aid is more often thought of as one arising under the Due Process Clause or the Eighth Amendment. See, e.g., DeShaney v. Winnebago County Dep’t of Soc. Servs., 489 U.S. 189, 200, 109 S.Ct. 998, 1005, 103 L.Ed.2d 249 (1989) (“[W]hen the State by the affirmative exercise of its power so restrains an individual’s liberty that it renders him unable to care *596for himself, and at the same time fails to provide for his basic human needs — e.g., food, clothing, shelter, medical care, and reasonable safety — it transgresses the substantive limits on state action set by the Eighth Amendment and the Due Process Clause.”). In fact, in their brief, the plaintiffs cite DeShaney and contend that the officers violated a constitutional duty by placing Mr. Phillips in a position so that he could not care for himself and then failing to provide for his medical needs, although counsel for the plaintiffs seemed to withdraw from this position to some degree at oral argument. In any event, the Due Process Clause is inapplicable in the present case. As we noted earlier, the Supreme Court has held that “all claims that law enforcement officers have used excessive force — deadly or not — in the course of an arrest, investigatory stop, or other ‘seizure’ of a free citizen should be analyzed under the Fourth Amendment and its ‘reasonableness’ standard, rather than under a ‘substantive due process’ approach.” Graham, 490 U.S. at 395, 109 S.Ct. at 1871 (emphasis in original). Although it is sometimes difficult to determine when an arrest has ended, an issue with which we have struggled,2 there is no question that, in this case, the complained-of police actions occurred during the course of Mr. Phillips’ seizure. Graham itself establishes as much. Like Mr. Phillips, the petitioner in Graham already had been handcuffed before the officers applied the physical force of which the petitioner complained. See 490 U.S. at 389-90, 109 S.Ct. at 1868. Indeed, similar to our case, one of the allegations of the Graham petitioner, a diabetic, was that the officers had ignored “pleas to get him some sugar.” Id. at 389, 109 S.Ct. at 1868. The Supreme Court nonetheless decided that the Fourth Amendment, not the Due Process Clause, was the proper standard by which to measure the petitioner’s claims. Therefore, although “[i]t is true that at some point after a person is arrested, the question whether his' continued confinement or prosecution is unconstitutional passes over from the Fourth Amendment to the due process clause (and after conviction to the Eighth Amendment’s cruel and unusual punishments clause ... ),” Jones v. City of Chicago, 856 F.2d 985, 994 (7th Cir.1988), that point was not crossed in this case, and the Fourth Amendment’s reasonableness standard remains the sole standard by which to measure the officers’ actions. That being said, although the officers’ actions here are not readily thought of as “force,” the Fourth Amendment requires that seizures be reasonable under all the circumstances; and we do think that it would be objectively unreasonable in certain circumstances to deny needed medical attention to an individual placed in custody who cannot help himself. Indeed, we have remarked before that the Fourth Amendment and the Due Process Clause impose similar duties in that both prohibit excessive force, though the duties apply at different times in the adversarial process and though the respective standards of liability may vary, primarily because the Due Process Clause contains a mental component. See Titran v. Ackman, 893 F.2d 145, 147-48 (7th Cir.1990); Wilkins v. May, 872 F.2d 190, 193 (7th Cir.1989), cert. denied, 493 U.S. 1026, 110 S.Ct. 733, 107 L.Ed.2d 752 (1990). Therefore, although we do not foreclose the possibility that a claim like the plaintiffs’ could be cognizable under the Fourth Amendment, we have determined that the officers in this case were objectively reasonable in them handling of Mr. Phillips’ medical situation.3
B.
Having decided that the officers did not violate the Constitution, we must con-*597elude that neither the City nor Police Chief Arreola can be held liable for Mr. Phillips’ death. See Thompson v. Boggs, 33 F.3d 847, 859 (7th Cir.1994), cert. denied, 514 U.S. 1063, 115 S.Ct. 1692, 131 L.Ed.2d 556 (1995). The City and Chief Arreola “were sued only because they were thought legally responsible for [the officers’] actions; if the latter inflicted no constitutional injury on [Mr. Phillips], it is inconceivable that [the former] could be liable to [the plaintiffs].” City of Los Angeles v. Heller, 475 U.S. 796, 799, 106 S.Ct. 1571, 1572, 89 L.Ed.2d 806 (1986) (per curiam). Similarly, “[i]f a person has suffered no constitutional injury at the hands of the individual police offieer[s], the fact that the departmental regulations might have authorized the use of constitutionally excessive force is quite beside the point.” Id. (emphasis omitted). Neither the City nor the police officers’ supervisor can be held liable on a failure to train theory or on a municipal policy theory absent a finding that the individual police officers are liable on the underlying substantive claim. See id.; Abbott v. City of Crocker, 30 F.3d 994, 998 (8th Cir.1994); Temkin v. Frederick County Comm’rs, 945 F.2d 716, 724 (4th Cir.1991), cert. denied, 502 U.S. 1095, 112 S.Ct. 1172, 117 L.Ed.2d 417 (1992).
Likewise, “[b]ecause we find that there was no constitutional violation, it is unnecessary to consider whether [the officers] w[ere] entitled to qualified immunity.” Kraushaar v. Flanigan, 45 F.3d 1040, 1049 n. 4 (7th Cir.1995) (citing Cornfield v. Consolidated High Sch. Dist. No. 230, 991 F.2d 1316, 1328 (7th Cir.1993) (Easterbrook, J., concurring)); accord Crowder v. True, 74 F.3d 812, 815 (7th Cir.1996) (per curiam). In this case, the officers’ actions did not violate any of Mr. Phillips’ constitutional rights; thus, there is no need to decide if those rights were clearly established at the time of the encounter. See Siegert v. Gilley, 500 U.S. 226, 111 S.Ct. 1789, 114 L.Ed.2d 277 (1991); Schlessinger v. Salimes, 100 F.3d 519, 523 (7th Cir.1996), cert. denied, — U.S. —, 117 S.Ct. 2481, 138 L.Ed.2d 990 (1997).
C.
Finally, we must address the district court’s decision to grant summary judgment on the plaintiffs’ state law claims, which include allegations of, for example, battery and wrongful death. The plaintiffs’ initial brief to this court contains no argument that the district court erred in its disposition of these claims. Rather, the plaintiffs first maintain that the district court should not have dismissed their state law claims in two pages of their reply brief. “[S]ince this argument was not clearly presented in [the plaintiffs’] initial brief to this Court, it could be deemed waived.” Arch of Illinois, Div. of Apogee Coal Corp. v. District 12, United Mine Workers of America, 85 F.3d 1289, 1294 (7th Cir.1996) (citing Wilson v. O’Leary, 895 F.2d 378, 384 (7th Cir.1990) (“All arguments for reversal must appear in the opening brief, so that the appellee may address them.”)). A necessary corollary to this principle is that “[arguments raised for the first time in the reply brief are waived.” Wildlife Express Corp. v. Carol Wright Sales, Inc., 18 F.3d 502, 508 n. 5 (7th Cir.1994); accord Giffin v. Summerlin, 78 F.3d 1227, 1231 (7th Cir.1996) (per curiam). In their reply brief, the plaintiffs purported to be replying to one paragraph at the end of the defendants’ brief in which the defendants stated that the district court had properly dismissed the plaintiffs’ state law claims. The defendants’ perfunctory paragraph remarking that they thought the district court correct in disposing of the state law claims is not enough to put those claims back in play on appeal. We recently explained that allowing an appellant to raise new arguments in reply “would be prejudicial not only to the appellee, but potentially to the appellant as well.” Parrillo v. Commercial Union Ins. Co., 85 F.3d 1245, 1250 (7th Cir.1996). If the plaintiffs wanted to press their state law claims on appeal, they should have done so in their initial brief to this court.
At any rate, Wisconsin law makes clear that the district court correctly granted summary judgment to the officers and the City on the plaintiffs’ state law claims. “ ‘The general rule acknowledged in Wisconsin is that a public officer or employee is immune from personal liability for injuries resulting from acts performed within the *598scope of the individual’s public office.’ ” Barillari v. City of Milwaukee, 194 Wis.2d 247, 533 N.W.2d 759, 763 (1995) (quoting C.L. v. Olson, 143 Wis.2d 701, 422 N.W.2d 614, 617 (1988)); see Wis. Stat. § 893.80(4). In Barillari the Supreme Court of Wisconsin explained the importance of giving immunity to law enforcement authorities for their official acts:
[W]e conclude that the nature of law enforcement requires moment-to-moment decision making and crisis management.... For these reasons, it is clear that law enforcement officials must retain the discretion to determine, at all times, how best to carry out their responsibilities.
Routinely, police face critical situations, many of which have the potential for violence. ... In the course of their work, police must often try to console and reassure people who are distraught and fearful. Faced with escalating violence, they must continuously use them discretion to set priorities and decide how best to handle specific incidents. Police officers must be free to perform their responsibilities, using their experience, training, and good judgment, without also fearing that they or them employer could be held liable for damages from them allegedly negligent discretionary decisions.
533 N.W.2d at 764-65. There are, however, three exceptions to the general rule of immunity: (1) “[A] public officer or employee does not enjoy immunity if he or she engages in conduct which is malicious, willful, or intentional”; (2) “a public officer or employee is not immune from liability if he or she negligently performs a ministerial duty”; and (3) a public officer “may also face liability when he or she is aware of a danger that is of such quality that the public officer’s duty to act becomes absolute, certain and imperative.” Id. at 763 (internal quotations and citations omitted).
The plaintiffs initially assert that the first exception would apply here if a jury were to find that the officers acted out of anger following the struggle with Mr. Phillips. Assuming that the exception would apply if the officers acted out of anger (which we do not decide), there is no evidence from which a reasonable jury could arrive at such a conclusion. The plaintiffs point to no evidence in the record that would support an inference that the officers were acting out of anger. Indeed, Kassin testified that the officers appeared to be trying to help Mr. Phillips. Moreover, the officers never acted violently toward Mr. Phillips; they did not hit him or use abusive or angry language during the encounter. Once Mr. Phillips’ condition was ascertained, they performed CPR and otherwise assisted the medical personnel in attempting to revive him. The record forecloses the possibility that the officers’ conduct was “malicious, willful, or intentional.” Barillari, 533 N.W.2d at 763.
Nor is there a colorable argument that the third exception could apply to deny immunity to the defendants. There is no evidence that placing a person in a prone position gives rise to dangers of such qualities that an absolute duty arises; nor is there any evidence that the officers knew of any such dangers. Similarly, the medical testimony confirms that Mr. Phillips’ pre-existing health problems, and the significance thereof, would not have been detectable by a lay person in the officers’ position. Likewise, there is nothing in the record to suggest that the officers realized that Mr. Phillips was having trouble getting sufficient oxygen (or had stopped breathing) and failed to act. Indeed, all the witnesses confirm that the officers acted immediately upon the discovery that Mr. Phillips was a pulseless non-breather.
Finally, the plaintiffs contend, without any citation to authority, that the officers may have been required to arrest Mr. Phillips because he had committed criminal acts, thereby making the arrest a ministerial duty that was performed negligently. As we noted above, “a public officer or employee is not immune from liability if he or she negligently performs a ministerial duty.” Id. This exception is “[t]he most generally recognized exception to the rule of immunity.” Lister v. Board of Regents of Univ. Wis. Sys., 72 Wis.2d 282, 240 N.W.2d 610, 621 (1976). Yet “[a] public officer’s duty is ministerial only when it is absolute, certain and imperative, involving merely the performance of a specific task when the law imposes, prescribes and defines the time, mode and occasion for its performance with such certainty that nothing *599remains for judgment or discretion.” Id. at 622. Although neither party cites the relevant cases, Wisconsin case law clearly forecloses the plaintiffs’ attempt to avail themselves of this exception; whether and how to arrest an individual involve discretion. In Sheridan v. City of Janesville, 164 Wis.2d 420, 474 N.W.2d 799 (Ct.App.1991), the plaintiff brought a negligence action against a municipality and two police officers for “injuries received while the police officers were performing ... sobriety tests and executing the arrest.” Id. at 801. The court stated that “a claim that the police officers were negligent in deciding whether or not to arrest Sheridan would clearly be proscribed by the holding in Lister.” Id. at 802 (internal quotation omitted). The court further held that the officers’ conduct in executing the arrest was discretionary in nature:
Their decisions regarding the manner in which the arrest was effectuated involved numerous applications of governing statutes and case law. They were required to determine whether Sheridan should be searched, handcuffed, subjected to force during execution of the arrest, and given a breathalyzer. These decisions involved “subjective evaluation of the law[.]”
Id. (citation omitted) (also holding that officers’ conduct could not fall within the “known present danger” exception to immunity). Similarly, in Sherry v. Salvo, 205 Wis.2d 14, 555 N.W.2d 402 (Ct.App.1996), the Court of Appeals of Wisconsin followed Sheridan and held that officers who injured a hospital patient during the course of restraining him were immune from liability: “[T]he ... officers’ acts in restraining Sherry were highly discretionary. They were reacting to highly charged and dynamic circumstances in pursuing and subduing Sherry, and were using their professional judgment as to how that task might best be accomplished.” Id. at 405. See also State v. Dekker, 112 Wis.2d 304, 332 N.W.2d 816 (Ct.App.1983) (affirming dismissal of criminal complaints alleging that police officers had intentionally failed to perform CPR on arrested individual because duty to render first aid, including CPR, requires police officer to exercise judgment).
The court in Sheridan also concluded, on immunity grounds, that the plaintiff had failed to state a claim against the City for inadequate training and supervision because “the [C]ity’s training and supervision of its police officers concerning conduct during arrests of disabled persons involves governmental discretion.” 474 N.W.2d at 803. “The City is not liable for the discretionary acts of the police department and its officers, acts for which immunity is provided under § 893.80(4), Stats.” Barillari, 533 N.W.2d at 765; accord Sherry, supra. Because Wisconsin law is clear that the defendants are entitled to immunity under state law, the district court did not abuse its discretion in deciding to retain jurisdiction over the supplemental state law claims. See Van Harken v. City of Chicago, 103 F.3d 1346, 1354 (7th Cir.), cert. denied, — U.S. —, 117 S.Ct. 1846, 137 L.Ed.2d 1049 (1997). Accordingly, we affirm the district court’s decision to grant summary judgment in favor of the defendants on the plaintiffs’ state law claims.
Conclusion
For the reasons given in the foregoing opinion, the judgment of the district court is affirmed.
AFFIRMED.
. The plaintiffs have submitted materials that document cases around the Nation in which individuals have died in police custody while restrained in a manner that limited their oxygen intake. Those encounters set forth in the materials, however, typically involve officers’ using choke holds and hog-ties or transporting an individual in a prone position. Also, additional circumstances, such as drug use or other physical problems, usually contributed to the deaths in the situations studied.
. See Reed v. City of Chicago, 77 F.3d 1049, 1051-52 (7th Cir.1996); Titran v. Ackman, 893 F.2d 145, 147 (7th Cir.1990); Wilkins v. May, 872 F.2d 190, 192-93 (7th Cir.1989), cert. denied, 493 U.S. 1026, 110 S.Ct. 733, 107 L.Ed.2d 752 (1990). See generally Mitchell W. Karsch, Note, "Excessive Force and the Fourth Amendment: When Does Seizure End?” 58 Fordham L.Rev. 823 (1990).
. Of course, even if the Due Process Clause had applied and DeShaney-like duties had arisen in this case, the plaintiffs could have recovered for any failure to provide medical care only if they could have established that the officers had been deliberately indifferent to Mr. Phillips' needs. See Salazar v. City of Chicago, 940 F.2d 233, 238 (7th Cir.1991); Anderson v. Gutschenritter, 836 F.2d 346, 348-49 (7th Cir.1988). As should be clear from our Fourth Amendment analysis, this record would not permit such a finding.