Dr. Gerald Einaugler appeals from an order of the United States District Court for the Eastern District of New York (Korman, J.), entered March 8, 1996, denying his petition for a writ of habeas corpus pursuant to 28 U.S.C. § 2254. Einaugler had been convicted, after a jury trial in the New York State Supreme Court, of reckless endangerment in violation of N.Y. Penal Law § 120.20 and of willful patient neglect in violation of N.Y. Public Health Law § 12-b. In his habeas petition challenging the conviction, Einaugler argued (1) that there was insufficient evidence at trial to establish the elements of reckless endangerment or patient neglect beyond a reasonable doubt, (2) that the admission of evidence that the allegedly neglected patient subsequently died violated Einaugler’s right to due process, and (3) that the laws under which Einaugler was convicted are unconstitutional as applied to him because they make criminal the exercise of medical judgment and because they provided inadequate notice that his actions were within their scope. He renews these arguments on appeal from the denial of his petition. Because we conclude that there was sufficient evidence to support Einaugler’s conviction, and because we find his other claims to be without merit, we affirm the order of the district court.
I. Background
The charges against Einaugler arose from his treatment of Alida Lamour, a patient at the Brooklyn Jewish Hospital Nursing Home (BJHNH), where Einaugler worked. On Friday, May 18, 1990, six days before she died, Lamour was returned to the nursing home after treatment for renal disease at the nearby Interfaith Hospital. At the nursing home, Einaugler, acting as attending physician, mistakenly ordered that a feeding solution be administered through Lamour’s kidney dialysis catheter. Early on Sunday morning, May 20, 1990, after the feeding solution had been administered for two days, a nurse noticed that Lamour was having difficulty breathing, that her abdomen was distended, and that she had vomited. The nurse, realizing the error, immediately attempted to drain the remaining feeding solution out of the patient’s peritoneal space through the catheter. At approximately 6 a.m. on Sunday morning, Einaugler was notified of his error and told of Lamour’s condition.
At trial, there was conflicting evidence about the events of Sunday after 6 a.m. and before 4:30 p.m. There is no doubt that Einaugler called Dr. Irving Dunn, the Chief of Nephrology at Interfaith Hospital, who had treated Lamour during her last hospital stay, and sought his advice about Lamour. Einaugler testified that Dunn told him that it did not seem like an emergency, and that Einaugler should send Lamour to the hospital on Monday for further treatment. Dunn testified that he directed Einaugler to hospitalize the patient. His testimony is ambigu*839ous as to whether he conveyed to Einaugler the importance of doing this immediately. Sometime after talking to Dunn, Einaugler wrote the following note in Lamour’s chart: “Spoke to Doctor Dunn, (Renal). Told to send patient over for evaluation to IMC [Interfaith] ER [Emergency Room].” Dunn testified that this note accurately summarized his morning conversation with Einaugler. Einaugler instead testified that the note, although “grammatically incorrect,” was written after Lamour had been sent to the hospital late Sunday afternoon and signified that he, Einaugler, had given instructions to send Lamour to the emergency room.
Later, between 11 a.m. and 2 p.m. on Sunday, Einaugler reported the mistake to Dr. Albert Khaski, the supervising physician at the nursing home. He told Khaski that Dunn had said that Lamour’s condition was not an emergency and could wait until Monday for hospitalization. Einaugler testified that Khaski agreed. Khaski, on the other hand, testified that he had disagreed and had instructed Einaugler to transfer Lamour to the hospital that day. At approximately 4:30 p.m., a nurse informed Einaugler that Lam-our was less responsive, unable to take food by mouth, and looked weak. Einaugler then ordered that Lamour be transferred to Interfaith Hospital, where testimony suggests she went essentially untreated until Monday morning. On Monday, Lamour received lavage to remove the remaining feeding solution from her peritoneal cavity and antibiotics to prevent infection. She nonetheless died four days later.
Einaugler was subsequently charged and eonvicted of reckless endangerment and willful neglect for delaying-Lamour’s hospitalization once he knew that to do so would create a serious risk of physical injury. He was sentenced to incarceration for fifty-two weekends. The Appellate Division of New York State Supreme Court affirmed his conviction, People v. Einaugler, 208 A.D.2d 946, 618 N.Y.S.2d 414 (1994), and the Court of Appeals denied his application for leave to appeal. People v. Einaugler, 84 N.Y.2d 1031, 623 N.Y.S.2d 187, 647 N.E.2d 459 (1995). Einaugler then petitioned for a writ of habeas corpus. Einaugler v. Supreme Court of the State of New York, 918 F.Supp. 619 (E.D.N.Y.1996). The district court denied his petition, but granted a certificate of probable cause, and stayed the judgment and conviction pending the disposition of Einaugler’s appeal.
II. Discussion
We review a district court’s denial of a petition for a writ of habeas corpus de novo. See Maldonado v. Scully, 86 F.3d 32, 35 (2d Cir.1996).
A. Sufficiency of the Evidence
The Due Process Clause of the Fourteenth Amendment prohibits conviction “except upon proof beyond a reasonable doubt of every fact necessary to constitute the crime with which [the defendant] is charged.” In re Winship, 397 U.S. 358, 364, 90 S.Ct. 1068, 1073, 25 L.Ed.2d 368 (1970). Consequently, a state prisoner “is entitled to habeas corpus relief if it is found that upon the record evidence adduced at the trial no rational trier of fact could have found proof of guilt beyond a reasonable doubt.” Jackson v. Virginia, 443 U.S. 307, 324, 99 S.Ct. 2781, 2791-92, 61 L.Ed.2d 560 (1979) (footnote omitted). Hence, we must consider whether, as a matter of federal law, there was sufficient evidence for a jury to find that the prosecution proved the substantive elements of the crime as defined by state law. See id. at 324 & n. 16, 99 S.Ct. at 2792 & n. 16; Green v. Abrams, 984 F.2d 41, 44-45 (2d Cir.1993).1 In evaluating whether the evidence is sufficient, we must “view[] the evidence in the *840light most favorable to the prosecution.” Jackson, 443 U.S. at 319, 99 S.Ct. at 2789.
Einaugler claims that the district court erred in finding that the trial evidence was sufficient to support his conviction. He argues that no reasonable jury could have found that he ignored the requisite standard of professional conduct in failing to send Lamour to the hospital before Sunday afternoon. We disagree. An appellant challenging the sufficiency of the evidence “bears a very heavy burden.” Quirama v. Michele, 983 F.2d 12, 14 (2d Cir.1993) (citation and internal quotation marks omitted). Einaugler has not met that burden, for, as is not infrequently the case where sufficiency of the evidence is challenged, this is a situation in which a jury could have found, and apparently did find, that “the combination of many small pieces of valid evidence — each of which [was] established ‘more probably than not’ ” was adequate, in sum, to establish the elements of each crime beyond a reasonable doubt. United States v. Martinez, 54 F.3d 1040, 1045 (2d Cir.1995) (Calabresi, concurring), cert. denied, — U.S. -, 116 S.Ct. 545, 133 L.Ed.2d 448 (1995).
The reckless endangerment charge required proof that Einaugler had recklessly engaged in conduct that created a substantial risk of serious physical injury. N.Y. Penal Law § 120.20. For Einaugler’s act to be reckless, he must have grossly deviated from a reasonable person’s standard of conduct and consciously disregarded a substantial and unjustifiable risk. See N.Y. Penal Law § 15.05 (defining “recklessly”).
Despite the fact that when Lamour was finally sent to the hospital she was not treated until the next day, a rational jury could have concluded from Dr. Dunn’s testimony and from Einaugler’s note on the patient’s chart: (1) that Einaugler called Dunn for instructions regarding Lamour’s treatment, (2) that Dunn told Einaugler to transfer Lamour to the hospital, (3) that Dunn conveyed to Einaugler the importance of doing this promptly, and (4) that Einaugler ignored that message. Although Dunn did not recall whether he explicitly informed Einaugler that the transfer should be immediate, a jury could read his testimony as being unequivocal in directing Einaugler to transfer Lam-our, and that this instruction conveyed immediacy. For example, he testified that he “directed ... Einaugler to have the patient admitted, period,” and agreed that Einaugler’s note in Lamour’s chart accurately summarized his order to Einaugler “to send Alida Lamour to the hospital that Sunday morning.” Einaugler’s note itself, since it clearly can be read to state that Einaugler was told by Dunn to sent the patient to Interfaith Hospital’s emergency room (and not simply to the hospital) lends important support to a finding that Dunn conveyed urgency.2 Dr. Khaski’s testimony provides some further basis for the conclusion that Einaugler was told that he should hospitalize Lamour in short order.
A jury could also have found that Einaugler understood that Dunn had directed him to hospitalize Lamour immediately, that he knew that Dunn’s instructions represented the requisite standard of medical care, and that he nevertheless failed to do as Dunn directed. A rational jury could, for example, have viewed the note as clear evidence that Einaugler both got the message that Dunn intended an immediate transfer, and that he recognized Dunn as an expert whose instructions indicated the appropriate treatment for Lamour. It follows that there is sufficient evidence to uphold a jury conclusion that Einaugler consciously deviated from what he knew to be the appropriate standard of care.
There was also sufficient evidence that the delay in hospitalizing Lamour created a substantial risk of serious physical injury and that the defendant was aware of that risk. See N.Y. Penal Law § 15.05; People v. Licitra, 47 N.Y.2d 554, 419 N.Y.S.2d 461, 393 N.E.2d 456 (1979). A “serious physical injury” is a “physical injury which creates a substantial risk of death, or which causes death, or which causes death or serious and *841protracted disfigurement, protracted impairment of health or protracted loss or impairment of the function of any bodily organ.” N.Y. Penal Law § 10.00(10).
Although there was no direct expert testimony that a delay would cause a risk of death, there was testimony that Lamour had peritonitis, that peritonitis must be treated, that peritonitis may cause fluid to enter the peritoneal cavity, that this process robs the blood stream of fluid and thus places a strain on the cardiovascular system, that peritonitis may be fatal especially to patients with cardiovascular problems or other medical conditions, and that Lamour suffered from cardiovascular disease and was otherwise in very poor health. Further testimony indicated that proper treatment would have involved lavage to remove the remaining feeding solution and intravenous fluid injection, and that such treatment could have prevented the consequences of fluid loss. Dunn himself, by analogizing Lamour’s situation to that of victims of the Chernobyl accident, strongly indicated that while there was no risk of Lamour dying immediately from the peritonitis, what had happened to her — if untreated — caused a substantial risk of death some time later.
Finally, trial evidence depicting Einaugler’s medical background and Einaugler’s own testimony on what he knew about peritonitis and its consequences provided a suffieient basis for a rational jury to conclude that Einaugler understood that the delay in hospitalizing Lamour would cause a substantial risk of death. This evidence, viewed as a whole, provided an adequate basis for a rational jury to conclude that Einaugler was guilty of reckless endangerment beyond a reasonable doubt.
Einaugler’s conviction for wilfully violating a public health law, pursuant to N.Y. Public Health Law § 12(b), by neglecting a patient in violation of N.Y. Public Health Law § 2803-d(7) is also supported by the evidence at trial. Under this law, a doctor neglects a patient when he or she “fail[s] to provide timely, consistent, safe, adequate, and appropriate services, treatment, and/or care to a patient.” N.Y.Comp. Codes R. & Regs., tit. 10 § 81.1[e] (1996). In order to violate the law wilfully, Einaugler must have known that his conduct was illegal. See People v. Coe, 71 N.Y.2d 852, 527 N.Y.S.2d 741, 743, 522 N.E.2d 1039 (Ct.App. 1988). For the reasons given above, the evidence at trial was sufficient for a rational jury to have concluded beyond a reasonable doubt that the appropriate standard of care required Einaugler to transfer the patient to the hospital once his mistake was discovered, and that Einaugler did not do so in a timely manner. Moreover, despite Einaugler’s testimony, a reasonable jury could have concluded that Einaugler was aware that a failure to transfer Lamour immediately violated the appropriate standard of care. Since at trial Einaugler stipulated that he knew that neglecting a patient was illegal, we hold, as did the district court, that there was sufficient evidence at trial for the jury to conclude beyond a reasonable doubt that Einaugler was guilty of willful patient neglect.
We must, as we have done above, evaluate a habeas petition that argues insufficiency of the evidence “with explicit reference to the substantive elements of the criminal offense as defined by state law.” Jackson, 443 U.S. at 324 n. 16, 99 S.Ct. at 2792 n. 16. While the state might have strengthened its case against Einaugler by presenting expert testimony directly stating that Einaugler’s delay in transferring Lam-our caused a substantial risk of death, New York law has never required such testimony in a criminal case of reckless endangerment. This court has recognized that New York civil tort actions for medical malpractice in New York cannot get to a jury without such expert evidence, see, e.g., Sitts v. United States, 811 F.2d 736, 739-41 (2d Cir.1987). But we did so in recognition of the fact that nearly one hundred years of New York eases specified such a requirement. See id. (citing cases); Robbins v. Nathan, 189 A.D. 827, 179 N.Y.S. 281 (1919) (citing earlier cases). In other words, the New York courts established this rule. We did not. There is no indication at all that a similar requirement exists in New York as to criminal reckless endangerment. (In fact, this ease provides evidence to the contrary since there was no such direct expert testimony in this case, and the New York courts affirmed *842the conviction.) Nor should we, absent such indications, extend the rule to criminal cases. While it may seem undesirable to differentiate between the forms of evidence required at criminal and civil trials involving standards of medical care, there may be grounds for doing so, such as the light burden of proof required to win a civil case, and the fact that scienter is needed to support a criminal conviction but not a civil malpractice verdict.
Habeas corpus review is narrow and constrained. We may only overturn a state conviction when that conviction was obtained in violation of a federal constitutional right. A state rule of evidence “does not run foul of the Fourteenth Amendment because another method may seem to our thinking to be fairer or wiser or to give a surer promise of protection to the prisoner at the bar.” Snyder v. Commonwealth of Massachusetts, 291 U.S. 97, 105, 54 S.Ct. 330, 332, 78 L.Ed. 674 (1934). It would go far beyond the proper role of this court in considering a writ of habeas corpus to determine that, by failing to recognize that New York law implicitly requires expert testimony to convict a criminal defendant of reckless endangerment or of wilful patient neglect in a medical case, three New York courts deprived Einaugler of his constitutional rights. To do so would be to use the federal constitution to micromanage the laws of New York State. Cf. Jackson, 443 U.S. at 324 n. 16, 99 S.Ct. at 2792 n. 16 (indicating that federal courts avoid “intrusions upon the power of the States to define criminal offenses” by reviewing habeas petitions that allege insufficiency of the evidence by referencing the “the substantive elements of the criminal offense as defined by state law”).
Although it would have violated Einaugler’s constitutional rights to be convicted without whatever proof is required by the laws of New York, we find no requirement of direct expert testimony in New York law, and hence we cannot conclude that a constitutional violation occurred in his case.3
B. Einaugler’s Other Claims
Einaugler’s arguments regarding the admission of evidence of the patient’s death and the constitutional invalidity of the statutes, as they were applied to him, can be dismissed summarily.
Einaugler argues that his conviction is unconstitutional as applied because the statutes under which he was convicted provided inadequate notice to him that his conduct might be criminal and because they punished him for exercising his medical judgment. A penal statute is void for vagueness unless it “gives the person of ordinary intelligence a reasonable opportunity to know what is prohibited and ... provides explicit standards for those who apply it.” United States v. Nadi, 996 F.2d 548, 550 (2d Cir.1993) (internal quotation marks, brackets, and citations omitted). Einaugler was convicted pursuant N.Y. Public Health Law § 12-b and N.Y. Penal Law § 120.20. The language of these statutes and the definitions to which they refer are not vague in that they are clearly meant to apply to situations such as this one. Moreover, the laws include a scienter requirement — that the endangerment be reckless and that the neglect be knowing— and the Supreme Court has stated “a scienter requirement may mitigate a law’s vagueness especially with respect to the adequacy of notice to the complainant that his conduct is proscribed.” Village of Hoffman Estates v. Flipside, Hoffman Estates, Inc., 455 U.S. 489, 499, 102 S.Ct. 1186, 1193, 71 L.Ed.2d 362 (1982) (footnote omitted). Nor was Einaugler punished under the statute simply for exercising reasonable medical judgment erroneously. See Colautti v. Franklin, 439 U.S. 379, 395-97, 99 S.Ct. 675, 685-87, 58 L.Ed.2d 596 (1979). To convict Einaugler, the jury was required to find, and did find, that he was aware that the patient required a course of treatment that he did not provide. Accordingly, Einaugler’s claims based on vagueness and arbitrariness of application are without merit.
*843Einaugler also contends that the district court erred in rejecting his argument that prosecution testimony mentioning Lam-our’s death and discussing its cause denied Einaugler a fair trial because such testimony was irrelevant to the crimes charged and was highly prejudicial. Even if we assumed that the testimony was erroneously admitted, and that this was an error of constitutional magnitude, we do not believe, considering the case as a whole, that the evidence of Lam-our’s death was so prejudicial as to create a reasonable probability that the jury would have given a different verdict had that evidence not been introduced. We thus have no basis to reverse on this ground.
III. Conclusion
We believe that a rational trier of fact could have found proof of Einaugler’s guilt beyond a reasonable doubt. We also hold that the statutes under which Einaugler was charged were not unconstitutional as applied to him, and or that he was not denied a fair trial by the introduction of evidence that he alleges was prejudicial. Accordingly, we affirm the order of the district court.
. We need not consider whether Einaugler would be entitled to habeas relief if New York state had convicted him on the basis of evidence that, though federally sufficient, was insufficient as a matter of state law, where state law imposes a higher standard for sufficiency than does federal law. This is so, first, because he failed to raise such a claim, and second, because New York applies the federal habeas standard described in Jackson to evaluate sufficiency of the evidence on appeal. See Liberta v. Kelly, 839 F.2d 77, 80 n. 1 (2d Cir.1988); Hawkins v. West, 706 F.2d 437, 440 (2d Cir.1983); People v. Contes, 60 N.Y.2d 620, 467 N.Y.S.2d 349, 350, 454 N.E.2d 932, 933 (1983).
. Einaugler, of course, contended that the note was a subsequent, ungrammatical, recording of his own decision to send Lamour to the hospital. On a challenge to the sufficiency of the evidence, however, we must take it as given that the jury chose not to believe Einaugler’s version of what the note signified, and instead read it to reflect an order by Dunn.
. A constitutional violation could still be found if the United States Constitution mandated that every conviction of a medical doctor for negligence requires expert testimony to establish the standard of care. But that is neither plausible nor seriously argued.