WILSON, Circuit Judge

Ellen Meltzer, U.S. Department of Justice, Criminal Division, James I. Pearce, U.S. Attorney General's Office, WASHINGTON, DC, Timothy J. Abraham, Kathleen Mary Salyer, MIAMI, FL, for Plaintiff–Appellee.

Ken Swartz, Law Office of Kenneth M. Swartz, MIAMI, FL, for Defendant–Appellant Doris Crabtree.

Aubrey Quinton Webb, Law Offices of Aubrey Webb, PA, CORAL GABLES, FL, for Defendant–Appellant Liliana Marks.

Joaquin Mendez, Jr., Joaquin Mendez, PA, MIAMI, FL, for Defendant–Appellant Roger Rousseau.

Richard Carroll Klugh, Jr., Law Offices of Richard C. Klugh, MIAMI, FL, for Angela Salafia.

Before WILSON and ROSENBAUM, Circuit Judges, and ROBRENO, District Judge.

WILSON, Circuit Judge:

This appeal involves multiple defendants convicted in an extensive Medicare fraud scheme. From 2004 to 2011, Health Care Solutions Network, Inc. (HCSN), an operator of mental health centers in Miami and North Carolina, billed Medicare for over $63M in fraudulent claims. Defendants Dr. Roger Rousseau, Doris Crabtree, Liliana Marks, and Angela Salafia are former employees of HCSN who were charged and convicted in connection with that fraud.

A jury convicted all the defendants of conspiracy to commit healthcare fraud, in violation of 18 U.S.C. § 1349, and also convicted Dr. Rousseau of two counts of healthcare fraud, in violation of 18 U.S.C. §§ 1347 and 2. They now appeal on multiple grounds, including double jeopardy, sufficiency of the evidence, and numerous procedural and evidentiary decisions made by the district court which they claim rendered the trial fundamentally unfair. Rousseau also appeals his Guidelines sentencing enhancements.

Upon thorough review and with the benefit of oral argument, we affirm on all issues.

I.

A. Background

HCSN was set up as a "partial hospitalization program" (PHP), which are designed to provide intensive psychiatric therapy to patients with "serious and acutely symptomatic mental illnesses." United States v. Willner , 795 F.3d 1297, 1302 (11th Cir. 2015). These programs serve as a bridge between restrictive inpatient care, such as a psychiatric hospital, and routine outpatient care. A PHP that complies with federal law and state coverage requirements may seek Medicare reimbursement for its services.

In accordance with Medicare standards, patients should be referred to a PHP by a psychiatrist or mental health specialist. Incentivizing such referrals through kickbacks violates federal law. 42 U.S.C. § 1320a-7b(b)(2)(A). Qualifying PHP patients must be admitted "under the supervision of a physician pursuant to an individualized, written plan of treatment." 42 U.S.C. § 1395x(ff). Treatment involves intensive, all-day individual and group therapy, which must not be "primarily recreational or diversionary." Id. The attending physician is responsible for managing each patient’s treatment, which must be "reasonable and necessary for the diagnosis or active treatment of the individual’s condition." Id.

While Medicare does not prescribe a fixed amount of time that patients should remain at a PHP, stays in excess of three months indicate that treatment is ineffective and no longer appropriate. Decisions to admit and discharge patients must be closely monitored by the attending physician.

A PHP seeking Medicare reimbursement must also comply with strict documentation requirements. At intake, medical personnel record patients’ history, symptoms, and medications. Once enrolled, a patient’s progress must be thoroughly charted, from the types of treatment provided to details of participation in specific therapy sessions. This both provides a medical record justifying the patient’s continued treatment and ensures that each patient is discharged—or readmitted—consistent with her medical needs.

B. The Fraud at HCSN

Such were not the practices at HCSN. From intake to discharge, HCSN organized its business around procuring, retaining, and readmitting patients to maximize billing potential, without respect to patients’ health needs. It then ensured patient files complied with Medicare coverage requirements by editing intake information, fabricating treatment plans, and falsifying therapy and treatment notes.

HCSN’s owner, Armando Gonzalez, orchestrated this fraud scheme, which involved coconspirators at every level of the company. Rousseau, the medical director at HCSN’s Miami locations, was the attending physician for over $30M of fraudulent billing to Medicare at HCSN-East. Crabtree, Marks, and Salafia were in charge of day-to-day therapy at HCSN-East, where they were also responsible for patient therapy notes.

The fraud began with patient recruitment. HCSN solicited patients from hospital employees and owners of assisted living facilities in exchange for cash kickbacks. Many of these patients did not qualify for PHP treatment. Upon intake, HCSN employees altered patient profiles in order to conceal disqualifying information, such as evidence that patients suffered from Alzheimer’s disease or dementia. HCSN treated patients for inordinately long periods of time—generally at least four months—and then "recycled" patients by discharging them and immediately admitting them to another HCSN location.

Patient notes and billing sheets were systematically altered to support Medicare claims and to survive Medicare audits. Therapists fabricated therapy notes for absent patients, falsified details from therapy sessions, and "cloned" notes by copy and pasting therapy notes, verbatim, from one patient’s file to another’s. "Ghost lists" of non-existent patients helped HCSN employees organize "ghost billing" of services that never took place.

This scheme spanned seven years, involved three HCSN locations, and amounted to over $63M in fraudulent claims.

C. Procedural History

HCSN’s owner, numerous employees, and beneficiaries of the illegal kickbacks were charged in connection with the healthcare fraud at HCSN. The defendants’ indictment charged them with the following: (1) all defendants with conspiracy to commit health care fraud, 18 U.S.C. § 1349 ; (2) Rousseau with two substantive counts of health care fraud, 18 U.S.C. §§ 1347 and 2; and (3) Crabtree, Marks, and Salafia with two counts of making false statements related to health care matters, 18 U.S.C. § 1035. Two trials ensued.

At the conclusion of the first trial, the jury acquitted Crabtree, Marks, and Salafia of the false statements counts but failed to reach a verdict on all other counts. At trial, the district court had granted the government’s request to deliver an instruction for Pinkerton liability with the false statements jury instruction. Once acquitted of the false statements charges, Crabtree, Marks, and Salafia moved for acquittal on the conspiracy count; they argued that the jury’s false statements acquittal necessarily entailed a finding that they were not a part of a conspiracy. The district court denied the motion for acquittal, holding that acquittal on the false statements counts did not foreclose retrial for conspiracy.

At the defendants’ second trial, a jury found all defendants guilty of conspiracy to commit healthcare fraud and found Rousseau guilty of both substantive counts of health care fraud. The district court applied a six-level enhancement to Rousseau’s guideline range—a two-level vulnerable victims enhancement and a four-level enhancement for organizing criminal activity involving more than five people. See U.S.S.G. §§ 3A1.1, 3B1.1. The district court sentenced Rousseau to 192 months in prison (100 months below the 292–365 month guideline range) and ordered him to pay $23M in restitution. It sentenced Crabtree, Marks, and Salafia to between five and six years’ imprisonment and ordered each to pay over $16M in restitution.

II.

The defendants now make multiple claims on appeal. We address them in four parts. First, we discuss Crabtree, Marks, and Salafia’s double jeopardy argument. Second, we determine whether there was sufficient evidence at trial to uphold the defendants’ convictions. Third, we tackle the trial-related claims: whether the district court erred (1) by admitting testimony and evidence about Medicare local coverage determinations, Medicare rules and regulations, PHP standard practices, and testimony regarding illegal kickbacks; (2) by dismissing a tardy juror; and (3) by instructing the jury on deliberate ignorance and aiding and abetting liability. Fourth, and finally, we determine whether the district court properly applied the two Guidelines enhancements to Rousseau’s sentence.

A. Double Jeopardy

Crabtree, Marks, and Salafia renew their collateral estoppel argument rejected at the first trial. They claim that because the jury acquitted them of the false statements counts—after the district court issued a Pinkerton liability instruction—the jury necessarily concluded as a factual matter that they were not a part of the healthcare fraud conspiracy at HCSN. And because that fact was essential to the conspiracy count, the Fifth Amendment barred the government from retrying them for conspiracy.

We review a district court’s collateral estoppel ruling de novo, and the "party asserting estoppel bears the burden of persuasion that the jury found the facts on which the defense of estoppel rests and that those facts bar another trial about them." United States v. Ohayon , 483 F.3d 1281, 1286 (11th Cir. 2007).

The Double Jeopardy Clause of the Fifth Amendment does not generally preclude the government from reprosecuting defendants on mistried counts. United States v. Shenberg , 89 F.3d 1461, 1478 (11th Cir. 1996). But the doctrine of collateral estoppel creates an exception to this rule "when an issue of ultimate fact has once been determined by a valid and final judgment," and when that issue constitutes an essential element of the mistried charge. Ashe v. Swenson , 397 U.S. 436, 443, 90 S.Ct. 1189, 1194, 25 L.Ed.2d 469 (1970). In such cases, the Fifth Amendment guarantee against double jeopardy prohibits the government from reprosecuting the defendant for the mistried charge. Id. at 445–46, 90 S.Ct. at 1195.

Thus, our collateral estoppel double jeopardy analysis involves two steps. "First, [the] court[ ] must examine the verdict and the record to see what facts, if any, were necessarily determined in the acquittal at the first trial. Second, the court must determine whether the previously determined facts constituted an essential element of the second offense." Ohayon , 483 F.3d at 1286 (internal quotation marks and citation omitted). An "essential element" is a factual, not legal, component of an offense. Id. at 1293. This is an objective inquiry which asks "whether a rational jury could have grounded its verdict upon an issue other than that which the defendant seeks to foreclose from consideration." Id. at 1286. Here, the defendants’ double jeopardy argument falls short, because acquittal on the false statements charges did not necessarily determine any factual issue essential to the healthcare fraud conspiracy charge. A rational jury could have acquitted without foreclosing the issue of whether the defendants were part of the healthcare fraud conspiracy at HCSN; therefore, retrial did not violate the Double Jeopardy Clause of the Fifth Amendment.

We begin with our first task: determining what facts were necessarily decided by the jury at the first trial. Id. Upon thorough review of the indictment, the arguments and evidence put forth at trial, and the jury instructions, we find that an acquittal on the false statements charges necessarily determined that Crabtree, Marks, and Salafia did not knowingly and willfully make materially false statements on six particular patient notes. Nor are they vicariously liable for that crime. But beyond that, "it is far from clear what facts the jury decided when it acquitted" the defendants on the false statements counts, and "[w]e will not speculate regarding the meaning of the jury’s verdict." United States v. Gil , 142 F.3d 1398, 1401 (11th Cir. 1998) (internal quotation marks omitted).

The jury was asked to determine the defendants’ liability with respect to six patient notes—but nothing more. From the indictment to closing arguments, the government limited the false statements charges to six specific notes. Indeed, the indictment spelled out the exact words on each note for which Crabtree, Marks, or Salafia was allegedly responsible, and these notes were the only evidence supporting the false statements counts at trial. This is illustrated by the closing arguments, where, for example, Doris Crabtree’s counsel argued:

Ladies and gentlemen, we have seen only two notes as far as Doris Crabtree is concerned.... Those are the notes on Counts 6 and 7 of the indictment.... [T]hose are the only two notes they have presented to you here in three weeks of trial that they accuse my client, Doris Crabtree, of fabricating. They want to destroy her life over two notes.

But the fact that the defendants did not fabricate these six notes was not essential to the healthcare fraud conspiracy count, on which the jury hung. That charge was much more sweeping, as demonstrated by the indictment and the record. The indictment alleged that Crabtree, Marks, and Salafia were involved with coconspirators in "submitting and causing the submission of false and fraudulent claims ... for services that were medically unnecessary, that were not eligible for Medicare and Medicaid reimbursement, and that were never provided," and that they "falsified, fabricated, altered, and caused the alteration, falsification, fabrication of HCSN medical records to support claims for PHP services that were not medically necessary and were not provided at HCSN-FL." Testimony at trial implicated Crabtree, Marks, and Salafia in these wider practices independently of the six patient therapy notes. Moreover, the government agreed not to use the six patient notes as evidence in the second trial—further confirmation that retrial did not violate the doctrine of collateral estoppel.

The Pinkerton instruction does not alter our conclusion. The defendants claim that an acquittal in light of Pinkerton necessarily determined that they were not a part of the healthcare fraud conspiracy at HCSN. But this is not so. The record suggests that a rational jury could have found that the government failed to meet its burden on any one of the elements for vicarious liability—especially given that the government failed to prove that the defendants themselves fabricated the six notes, which was its central theory at trial. The defendants have not met their burden of proving by convincing and competent evidence that the acquittal necessarily determined any factual issue essential to the mistried charge. See United States v. Hogue , 812 F.2d 1568, 1578 (11th Cir. 1987).

This distinguishes the defendants’ case from Ohayon and Larkin , on which they rely. In Ohayon , the defendant was caught transporting ecstasy inside duffle bags. 483 F.3d at 1282. At trial, the jury acquitted him of an intent to distribute charge and hung on a conspiracy charge. Id. But essential to both charges was the same mens rea issue: did the defendant know what was in the duffle bags he was transporting? Indeed, we found that "[t]here was no other factual issue" presented to the jury. Id. at 1287. Accordingly, when the jury acquitted on one count, it necessarily determined that essential issue (in the negative), which foreclosed the government from relitigating it in another trial for conspiracy.

But in this case, there was no "single rationally conceivable issue in dispute before the jury" when it considered the false statements charges. Cf. Ashe , 397 U.S. at 445, 90 S.Ct. at 1195. The Pinkerton instruction split the liability question into two and multiplied the grounds upon which a verdict could rest. Thus, besides the fact that the defendants did not make false statements on the six patient notes, we cannot say from this record that any other factual determination necessarily proceeded from the acquittal.

The defendants’ reliance upon the beleaguered United States v. Larkin is also misplaced. 605 F.2d 1360, 1370–71 (5th Cir. 1979), withdrawn in part on reh’g , 611 F.2d 585 (5th Cir. 1980), overruled in part by Yeager v. United States , 557 U.S. 110, 116–117, 129 S.Ct. 2360, 2365, 174 L.Ed.2d 78 (2009). In Larkin , part of which remains binding precedent, the jury acquitted the defendant on six substantive counts—tried exclusively on a theory of vicarious liability under Pinkerton —but failed to reach a verdict on the conspiracy charge. Id. at 1362–63. Importantly, and in contrast to this case, the six substantive charges were the sole crimes underlying the conspiracy charge. Id. at 1363. It followed that, because the substantive charges and the conspiracy charge wholly overlapped, acquitting the defendant of conspiracy liability on all of the substantive charges necessarily foreclosed the issue of whether the defendant was guilty of conspiracy.

But in this case, as discussed above, the six notes at issue in the false statements charges were not the sole evidence supporting the healthcare fraud conspiracy charge. Therefore, a determination that the defendants were not vicariously liable for the false statements crimes did not foreclose the issue of whether the defendants joined the healthcare fraud conspiracy at HCSN. Larkin , or what’s left of it, does not support the defendants’ position.

We reiterate that our task is to determine, objectively and "in a practical frame," what facts a rational jury must necessarily have decided in producing a verdict. Ashe , 397 U.S. at 444, 90 S.Ct. at 1194. We are not to speculate as to the actual source of a verdict, Gil , 142 F.3d at 1401, nor to foreclose from relitigation all factual issues which theoretically could have been involved in an acquittal. At the first trial, an acquittal on the false statements charges determined only that Crabtree, Marks, and Salafia did not make materially false statements on six particular patient notes, and that they were not vicariously liable for that crime. Because neither of those issues was essential to the healthcare fraud conspiracy count, the government was not estopped from retrying the defendants for that charge. We therefore affirm the district court’s denial of the motion for acquittal.

B. Sufficiency of the Evidence

Next, the defendants contest the sufficiency of the evidence to support their convictions at the second trial. According to Crabtree, Marks, and Salafia, the evidence against them was primarily circumstantial and demonstrated only that they were negligent and careless—but did not show that they knowingly and voluntarily joined a conspiracy to defraud Medicare. Rousseau similarly argues that while he irresponsibly abdicated his role as medical director, his only "sin" was that he "naively trusted others." He did not know that the documents he signed—or authorized others to sign on his behalf—were falsified.

We review whether there is sufficient evidence to support the jury’s guilty verdicts de novo, looking at the evidence "in the light most favorable to the government and resolving all reasonable inferences and credibility evaluations in favor of the verdict." United States v. Moran , 778 F.3d 942, 958 (11th Cir. 2015). "If there is a lack of substantial evidence, viewed in the Government’s favor, from which a reasonable factfinder could find guilt beyond a reasonable doubt, the conviction must be reversed." United States v. Willner , 795 F.3d 1297, 1307 (11th Cir. 2015). But we will not overturn a jury’s verdict "if any reasonable construction of the evidence would have allowed the jury to find the defendant guilty beyond a reasonable doubt." United States v. Wilson , 788 F.3d 1298, 1308 (11th Cir. 2015). A conviction for conspiracy to commit health care fraud in violation of 18 U.S.C. § 1349 requires proof beyond a reasonable doubt "(1) that a conspiracy existed; (2) that the defendant knew of it; and (3) that the defendant, with knowledge, voluntarily joined it." United States v. Vernon , 723 F.3d 1234, 1273 (11th Cir. 2013). "The very nature of conspiracy frequently requires that the existence of an agreement be proved by inferences from the conduct of the alleged participants or from circumstantial evidence of a scheme." United States v. Molina , 443 F.3d 824, 828 (11th Cir. 2006). The government need only prove that the defendant knew of the "essential nature" of the conspiracy, and "[w]e will affirm a conspiracy conviction when the circumstances surrounding a person’s presence at the scene of conspiratorial activity are so obvious that knowledge of its character can fairly be attributed to him." Vernon , 723 F.3d at 1273–74 (internal quotation marks omitted). The government can show that a defendant voluntarily joined a conspiracy "through proof of surrounding circumstances such as acts committed by the defendant which furthered the purpose of the conspiracy." Id. at 1274. "[A] defendant can be convicted of conspiracy even if his or her participation in the scheme is ‘slight’ by comparison to the actions of other coconspirators." Id. at 1273 (alteration adopted).

The health care fraud statute, 18 U.S.C. § 1347, reads in pertinent part that:

Whoever knowingly and willfully executes, or attempts to execute, a scheme or artifice—(1) to defraud any health care benefit program; or (2) to obtain, by means of false or fraudulent pretenses, representations, or promises, any of the money or property owned by, or under the custody or control of, any health care benefit program,

in connection with the delivery of or payment for health care benefits, items, or services, shall be fined under this title or imprisoned not more than 10 years, or both.

18 U.S.C. § 1347(a). Furthermore, "the defendant must be shown to have known that the claims submitted were, in fact, false." United States v. Medina , 485 F.3d 1291, 1297 (11th Cir. 2007).

1. Crabtree, Marks, and Salafia’s Conspiracy Convictions

We find sufficient evidence in the trial record for a reasonable jury to conclude that Crabtree, Marks, and Salafia had knowledge of the conspiracy at HCSN. Furthermore, the extensive "proof of surrounding circumstances" at HCSN coupled with the numerous, continuous "acts committed by the defendant[s] which furthered the purpose of the conspiracy" over a number of years lead us to conclude that a reasonable factfinder could determine that Crabtree, Marks, and Salafia voluntarily joined the conspiracy. Vernon , 723 F.3d at 1274. Accordingly, we must affirm their convictions.

The government put forth considerable evidence that Crabtree, Marks, and Salafia were directly aware of the "essential nature" of the conspiracy and that the circumstances at HCSN were "so obvious that knowledge of [the fraud’s] character can fairly be attributed to [them]." Id. at 1273–74. Multiple former-employees testified that Crabtree, Marks, and Salafia complied with their requests to doctor patient notes so that they would pass Medicare review. Furthermore, witnesses testified that kickback payments to assisted living facilities were routinely discussed around the office, and that Crabtree, Marks, and Salafia were present for these conversations. Numerous witnesses spoke of the overwhelming evidence that patients were unqualified for PHP treatment: that it was obvious, and widely observed, that patients at HCSN suffered from Alzheimer’s, dementia, autism, and forms of mental retardation that rendered treatment ineffective; that this was evidenced, for example, by patients relieving themselves around the office and by patients who were unable to engage in group therapy sessions; and that Crabtree, Marks, and Salafia were involved in multiple conversations about the unsuitability of HCSN’s patients for PHP treatment. One former-employee put it simply: "everybody was aware of the fraud."

Likewise, a reasonable jury could have found that Crabtree, Marks, and Salafia voluntarily joined the conspiracy, given the substantial evidence of their role in furthering the fraud. The government put forth evidence that Crabtree, Marks, and Salafia complied with requests to alter and fabricate notes for billing and Medicare auditing purposes; that they created therapy session notes for absent patients; that they routinely copied and pasted notes in patients’ files; and that each therapist misrepresented the therapy that patients received when, for example, they played movies (about once per week) during therapy sessions, or when patients were late or absent but notes indicated that they participated fully.

Ms. Dana Gonzalaz, who was in charge of ensuring that HCSN’s patient files and billing would pass Medicare review, testified that she asked Crabtree, Marks, and Salafia to retroactively edit patient notes to make them Medicare compliant. This included adding specific fabricated quotations and changing symptoms to give the impression that patients were progressing in accordance with their treatment plans. She testified that Crabtree, Marks, and Salafia would take the notes home, make the changes, and then sign off on them.

Additionally, Ms. Gema Pampin, a program coordinator at HCSN, testified that Crabtree, Marks, and Salafia complied with her requests to doctor patient notes so that HCSN would not get caught by Medicare. According to Pampin, Crabtree, Marks, and Salafia never refused to make the requested alterations and sign the notes.

And Ms. Lisset Palmero, the office manager at HCSN-East, testified that Crabtree, Marks, and Salafia fabricated therapy notes for absent patients. At trial, the government presented numerous patient therapy notes alongside a billing sheet, and Ms. Palmero identified (1) how each therapy note corresponded to a date that the patient was absent from HCSN and (2) how either Crabtree, Marks, or Salafia’s signature was affixed to each fabricated note.

In conclusion, viewing the evidence in a light most favorable to the government and making all reasonable credibility determinations in favor of the verdict, we find that the record provides sufficient evidence to support Crabtree, Marks, and Salafia’s healthcare fraud conspiracy convictions.

2. Dr. Rousseau’s Conspiracy and Fraud Convictions

We also find that there was sufficient evidence for a reasonable jury to conclude that Rousseau was knowingly and voluntarily involved in the conspiracy and knowingly signed false medical claims in order to defraud Medicare.

Rousseau was crucial to the fraud scheme at HCSN, and there was considerable evidence that he was aware of this. He signed off on 96% of the fraudulent Medicare claims that came out of HCSN-East, which amounted to over $30M in false claims. He was paid almost a million dollars, much of which came in lump-sum payments issued from clandestine bank accounts, to admittedly sign whatever medical documents were put in front of him. He appeared at HCSN once every couple of weeks, for a couple of hours, simply for this purpose. He rarely, if ever, met with patients or reviewed their medical needs, nor did he prescribe the individual treatment plans that he signed off on. And he authorized use of his signature stamp without permission and without review, often when he was present at the office, so that the fraudulent documents could be signed more quickly.

Furthermore, numerous witnesses testified directly to Rousseau’s knowledge of HCSN’s healthcare fraud conspiracy. He was part of conversations where employees were instructed to delete references to Alzheimer’s and dementia medications from patient profiles in order to make them Medicare compliant. He was aware that HCSN was recycling patients to increase billings. He introduced HCSN’s owner to a hospital employee who was then paid illegal kickbacks for recommending patients to HCSN, and who testified at trial that Rousseau followed up with him to see whether HCSN’s owner was "taking care of" him. Rousseau participated in a conversation over the "bones in the closet, the evil secrets of [HCSN]."

HCSN’s owner and multiple coconspirators testified against Rousseau directly and testified to the overwhelming circumstantial evidence of fraud at HCSN. Unlike in Willner , 795 F.3d 1297, upon which Rousseau relies, testimony at trial directly linked him to the fraud benefits, illegal kickbacks, patient recycling, doctored patient files, and the countless fraudulent medical claims that came out of HCSN, all of which he was ultimately accountable for as the attending physician. We conclude that a reasonable factfinder had sufficient evidence to convict Rousseau on all counts, and therefore affirm his convictions.

C. Trial Claims

1. Admission of Evidence and Testimony

The defendants first argue that the district court erred in admitting testimony about Medicare local coverage determinations and Medicare rules and regulations regarding PHPs, which, they claim, confused the jury and effectively lowered the burden of proof for conviction. Rousseau also argues that trial testimony concerning kickbacks, especially expert testimony about the Stark Law, was irrelevant and unduly prejudicial. The cumulative effect of these errors, defendants argue, resulted in a fundamentally unfair trial.

We review the admission of witness testimony, both lay and expert, for abuse of discretion. Moran , 778 F.3d at 958 ; United States v. Frazier , 387 F.3d 1244, 1258 (11th Cir. 2004). We review all evidentiary decisions for abuse of discretion. United States v. Brown , 415 F.3d 1257, 1264–65 (11th Cir. 2005). An abuse of discretion can occur where the district court "applies the wrong law, follows the wrong procedure, bases its decision on clearly erroneous facts, or commits a clear error in judgment." Id. at 1266.

The district court did not abuse its discretion in allowing the aforementioned evidence and testimony at trial, and any error was harmless.

First, testimony regarding Medicare coverage determinations was relevant to the government’s theory of motive: that HCSN employees altered records in order to get reimbursed by (or survive audits by) Medicare. Moreover, the district court issued limiting instructions both before experts testified regarding local coverage determinations and again before charging the jury. Specifically, the district court emphasized that violating Medicare rules and regulations was not a crime, but that it "may be relevant in determining whether a defendant acted with criminal intent ... to defraud Medicare." Dr. Elizabeth Crocco’s testimony, which explained a physician’s responsibility to patients in a PHP, also illumined how far afield HCSN’s practices were from normal, which was relevant to the knowledge element of the conspiracy and fraud charges.

Likewise, evidence regarding illegal kickbacks was directly relevant to Rousseau’s knowledge of and involvement in the conspiracy. And while it may have been prejudicial to allow Stephen Quindoza, an expert who formerly worked for a Medicare contractor, to testify that a doctor benefiting from kickbacks would be violating the Stark Law, any error was harmless given the court’s very clear limiting instructions and the overwhelming evidence presented against Rousseau at trial. See Willner , 795 F.3d at 1321.

Finally, given that there was, at most, one harmless trial error, the defendants’ cumulative error argument fails. United States v. House , 684 F.3d 1173, 1210–11 (11th Cir. 2012).

2. Juror Dismissal

Next, we consider the defendants’ argument that the district court erred when it dismissed a juror for being late during closing arguments. The defendants argue that there was no reasonable basis for the dismissal, and that dismissing the juror ran afoul of Federal Rule of Criminal Procedure 24(c) and the Sixth Amendment.

We review whether a district court improperly dismissed a juror for abuse of discretion. Moran , 778 F.3d at 958. A court may substitute an alternate juror for any juror who is "unable to perform" or is "disqualified from performing" his duties. Fed. R. Crim. P. 24(c). "The decision to remove a juror and replace him with an alternate is entrusted to the sound discretion of the trial judge whenever facts are presented which convince the trial judge that the juror’s ability to perform his duty as a juror is impaired." United States v. Fajardo , 787 F.2d 1523, 1525 (11th Cir. 1986) (internal quotation marks omitted). This discretion should not be disturbed absent a showing of bias or prejudice to the defendant, which includes discharge of a juror without factual support, or for a legally irrelevant reason. Id. "The trial judge does not need a defendant's consent to replace a juror with an alternate before the jury retires; all that is required is a reasonable cause for the replacement." Id. at 1526.

The district court had a reasonable ground for replacing the juror, who failed to arrive for court and who had already delayed closing arguments by fifteen minutes at the time she was replaced. This was clearly disruptive to the trial proceedings. The judge, noting that the juror had been late on several other occasions, refused to delay closing arguments an additional thirty minutes to wait for the juror, as requested by defense counsel.

We have noted that a juror’s absence "manifestly interferes with the prompt trial of a case. Hence when a juror is absent from court for a period sufficiently long to interfere with the reasonable dispatch of business there may be a ‘sound’ basis for his dismissal." United States v. Rodriguez , 573 F.2d 330, 332 (5th Cir. 1978). The defendants argue that the dismissal prejudiced them because they no longer had their originally selected jury. But "[e]very replacement involves a change in the jury’s composition," id. at 333, and it is therefore up to the defendants to demonstrate that the district court’s dismissal lacked a sound factual basis or was legally irrelevant—which they fail to do here. Accordingly, we conclude that the district court did not abuse its discretion in replacing the tardy juror with an alternate.

3. Jury Instructions

The defendants’ final trial-related claim is that the district court improperly charged the jury. Specifically, the defendants argue that the district court erred in delivering a deliberate ignorance instruction, which is only appropriate when there is evidence that a defendant purposefully avoided learning the truth to shield himself from prosecution; no such evidence was presented against defendants. Furthermore, in delivering this instruction, the district court used an analogy from the drug context, which the defendants allege confused the jury.

Marks also argues that the court erred by delivering an aiding and abetting instruction on the conspiracy count.

"We apply a deferential standard of review to a trial court’s jury instructions ... [and] will only reverse if we are left with a substantial and eradicable doubt as to whether the jury was properly guided in its deliberations." United States v. Steed , 548 F.3d 961, 977 (11th Cir. 2008) (per curiam). "The district court has broad discretion in formulating a jury charge as long as the charge as a whole is a correct statement of the law." United States v. Perez-Tosta , 36 F.3d 1552, 1564 (11th Cir. 1994).

"[A] deliberate ignorance instruction is warranted only when the facts support the inference that the defendant was aware of a high probability of the existence of the fact in question and purposely contrived to avoid learning all of the facts in order to have a defense in the event of a subsequent prosecution." Steed , 548 F.3d at 977 (internal quotation marks omitted). "We have held, however, that instructing the jury on deliberate ignorance is harmless error where the jury was also instructed and could have convicted on an alternative, sufficiently supported theory of actual knowledge." Id.

First, whether or not a deliberate ignorance instruction was appropriate here, the jury was also instructed on—and, as we have shown above, could have convicted on—a sufficiently supported theory of actual knowledge. Accordingly, any error in delivering the deliberate ignorance instruction was harmless. Id.

Second, the inclusion of the drug example was not reversible error. It was an accurate illustration of the legal principle, and no defendant objected to it at trial. The district court carefully delivered the Eleventh Circuit pattern instructions and even included, at the defendants’ request, an instruction from Willner , 795 F.3d at 1315, which clarified that deliberate ignorance can only establish the knowledge prong of the conspiracy charge.

And last, Marks is simply mistaken regarding the aiding and abetting instruction. The district court did not give an aiding and abetting instruction with the conspiracy count; it delivered it with the healthcare fraud charge only, where it was undisputedly appropriate.

In sum, none of the district court’s trial decisions warrant reversal.

D. Rousseau’s Sentencing Enhancements

Finally, we address Rousseau’s sentencing appeal. He first argues that the court erred in applying a U.S.S.G. Section 3B1.1 organizer or leader enhancement, because there was no evidence that he managed, directed, or controlled other participants in the fraud scheme. Indeed, Rousseau contends, he hardly showed up at all. All he contributed was his signature; he was medical director in name only. Further, Rousseau argues that the district court erred in applying the U.S.S.G. Section 3A1.1 vulnerable victims enhancement, because he did not personally and specifically target victims due to their vulnerability.

"We review a district court’s determination that a defendant is subject to a Section 3B1.1 role enhancement as an organizer or leader for clear error." United States v. Martinez , 584 F.3d 1022, 1025 (11th Cir. 2009). "We review de novo the district court’s application of the vulnerable victim enhancement, but we give due deference to the district court’s determination that a victim was vulnerable, as this is a factual finding." United States v. Birge , 830 F.3d 1229, 1231 (11th Cir. 2016) (internal quotation marks omitted).

The Sentencing Guidelines call for a four-level sentencing enhancement when the "defendant was an organizer or leader of a criminal activity that involved five or more participants or was otherwise extensive." U.S.S.G. § 3B1.1(a). The Guidelines comments suggest seven factors to consider in making this determination:

(1) exercise of decision making authority, (2) the nature of participation in the commission of the offense, (3) the recruitment of accomplices, (4) the claimed right to a larger share of the fruits of the crime, (5) the degree of participation in planning or organizing the offense, (6) the nature and scope of the illegal activity, and (7) the degree of control and authority exercised over others.

See Martinez , 584 F.3d at 1026.

In addition, the Guidelines require a two-level sentencing increase if the defendant "knew or should have known that a victim of the offense was a vulnerable victim." U.S.S.G. § 3A1.1(b)(1). A vulnerable victim is "a victim of the offense of conviction ... who is unusually vulnerable due to age, physical or mental condition, or who is otherwise particularly susceptible to the criminal conduct." Id. § 3A1.1 cmt. n.2; Birge , 830 F.3d at 1231–33.

Here, the district court did not clearly err in applying the organizer or leader enhancement to Rousseau’s sentence. While the evidence does not demonstrate that Rousseau closely managed and controlled operations at HCSN, he, as medical director, was in a pivotal position of management authority that enabled the fraud to succeed. Additionally, the government put on evidence that Rousseau was involved in conversations where he directed employees to remove certain medications from patient files, and he clearly authorized his subordinates to use his signature toward fraud. Rousseau also profited more than any other person convicted in the fraud scheme other than the owner himself. Taking the Section 3B1.1 factors into consideration, we cannot conclude that the district court clearly erred in applying the organizer or leader enhancement.

There is also sufficient evidence supporting the Section 3A1.1 vulnerable victims enhancement, which we must review de novo. Medicare was clearly not the sole victim of HCSN’s fraud. Numerous patients—many of whom did not qualify for PHP treatment—received inappropriate and inadequate treatment at HCSN, whose entire business was caring for elderly patients with "serious and acutely symptomatic mental illnesses." Willner , 795 F.3d at 1302.

More than anyone else at HCSN, Rousseau was responsible for these patients: for reviewing their medical needs and qualifications for admission; for developing an individualized treatment plan for their conditions; for overseeing the implementation of treatment plans; and for monitoring each patient’s progress and eligibility for release. He admittedly abdicated all of that responsibility by signing "whatever medical documents they put in front of him," in furtherance of fraud, without regard to the underlying implications for patients’ health and well-being. We affirm the application of the vulnerable victims enhancement to his sentence.

III.

In conclusion, we hold that the defendants’ retrial did not violate the Fifth Amendment guarantee against double jeopardy; that the district court did not commit any reversible error at trial in its evidentiary and procedural decisions, in replacing a tardy juror, and in delivering the jury instructions; and we affirm the application of the organizer or leader and vulnerable victims enhancements to Rousseau’s sentence.

AFFIRMED.