Opinion by Judge SCHROEDER; Concurrence by Judge SNEED.
SCHROEDER, Circuit JudgePlaintiffs challenge the constitutionality of an Arizona statute that criminalizes any medical “experimentation” or “investigation” involving fetal tissue from induced abortions unless necessary to perform a “routine pathological examination” or to diagnose a maternal or fetal condition that prompted the abortion. The plaintiffs include individuals suffering from Parkinson’s disease who because of the statute are unable in Arizona to receive transplants of fetal brain tissue that many medical experts believe hold out promise for eventual amelioration or treatment of the disease. Plaintiffs also include doctors in Arizona who fear possible criminal prosecution if they provide services to their patients that the doctors would like to provide.
The district court held on summary judgment that the statutes are unconstitutionally vague, and permanently enjoined their enforcement. It did not reach various other theories presented in plaintiffs’ complaint for invalidation of the statute. In so ruling the district court followed the holdings of three other circuits that considered similar statutes and held them all unconstitutionally vague. See Jane L. v. Bangerter, 61 F.3d 1493, 1499-1502 (10th Cir.1995), Rev’d and remanded on other grounds sub. norm., Leavitt v. Jane L., 518 U.S. 137, 116 S.Ct. 2068, 135 L.Ed.2d 443 (1996); Margaret S. v. Edwards, 794 F.2d 994, 998-99 (5th Cir.1986); Lifchez v. Hartigan, 735 F.Supp. 1361, 1363-76 (N.D.Ill.), aff'd mem., 914 F.2d 260 (7th Cir.1990). In this appeal by the state, we affirm the district court holding. Its decision is published at 71 F.Supp.2d 1015 (D.Ariz.1999). We do not repeat the procedural background.
The principal statute with which we are concerned is A.R.S. § 36-2302, subpart (A). It provides:
A person shall not knowingly use any human fetus or embryo, living or dead, or any parts, organs, or fluids of any such fetus or embryo resulting from an induced abortion in any manner for any medical experimentation or scientific or medical investigation purposes except as is strictly necessary to diagnose a disease or condition in the mother of the fetus or embryo and only if the abortion was performed because of such disease or condition.
Section 36-2302, subpart (C) provides an exception:
This section shall not prohibit any routine pathological examinations conducted by a medical examiner or hospital laboratory provided such pathological examination is not a part of or in any way related to any medical or scientific experimentation.
Thus the statute does not outlaw all use of fetal tissue derived from induced abortions. Instead it generally outlaws the use of such tissue for experimentation, subject to certain exceptions.
Persons violating Section 36-2302 commit a class 5 felony, a crime punishable by one-and-a-half years in prison, and face fines up to $150,000, see A.R.S. § 36-2303. Doctors found to have violated the statute also face censure, probation, suspension of license, revocation of license, or any combination of these. See A.R.S. §§ 13-701, 13-801, 32-1451, 32-1844.
In their complaint and supporting affidavits and depositions, the plaintiff physicians explain the types of procedures involving the use of fetal tissue that they *1011would use, were it not for the statute. They believe these procedures would fulfill their obligations to promote the health of their patients, and would also advance medical knowledge. Dr. Snider, one of the plaintiffs in this case, stated in his deposition that the statute prevented him from .prescribing and managing a course of treatment for his Parkinson’s disease patients that includes fetal tissue transplantation. Another plaintiff, Dr. Melcher, submitted an affidavit indicating that fetal tissue transplantation holds considerable promise for some of his Parkinson’s disease patients.
Fetal tissue is also useful in diagnosing and testing for fertility problems. One of the plaintiff physicians who- specializes in fertility treatments, Dr. Tamis, was the target of a potentially criminal investigation some years ago when he endeavored to study the effects on the fetus of a drug ingested by pregnant women before an induced abortion was performed. The study was to determine whether the drug passed through the placental wall. Although the state eventually dismissed the grand jury subpoenas issued to Dr. Tamis, he is still uncertain about the proper interpretation of the statute.
Other physicians and expert witnesses explain that many established treatments for illness have developed from fetal research and experimentation, including the polio vaccine. They point out the difficulties of knowing at what stage or point in time “experiments” become recognized as “treatment.” They also point out that the tenns “investigation” and “routine examination” are fundamentally ambiguous. In particular, the experts highlight doctors’ lack of consensus about what procedures are purely experimental. In the view of one expert submitted to the district court, virtually every procedure with a therapeutic objective is experimental to some extent.
The due process clause of the Fourteenth Amendment guarantees individuals the right to fair notice of whether their conduct is prohibited by law. Co-lautti v. Franklin, 439 U.S. 379, 390-91, 99 S.Ct. 675, 58 L.Ed.2d 596 (1979), citing United States v. Harass, 347 U.S. 612, 617, 74 S.Ct. 808, 98 L.Ed. 989 (1954). Although only constructive rather than actual notice is required, individuals must be given a reasonable opportunity to discern whether their conduct is proscribed so they can choose whether or not to comply with the law. Giaccio v. Pennsylvania, 382 U.S. 399, 402-03, 86 S.Ct. 518, 15 L.Ed.2d 447 (1966). Statutes need not be written with “mathematical” precision, nor can they be thus written. Grayned v. City of Rockford, 408 U.S. 104, 108-09, 92 S.Ct. 2294, 33 L.Ed.2d 222 (1972). But they must be intelligible, defining a “core” of proscribed conduct that allows people to understand whether their actions will result in adverse consequences. Planned Parenthood v. Arizona, 718 F.2d 938, 947 (9th Cir.1983)(holding that a statute is void for vagueness if persons of common intelligence must necessarily guess at its meaning).
If a statute subjects transgressors to criminal penalties, as this one does, vagueness review is even more exacting. See Kolender v. Lawson, 461 U.S. 352, 357, 103 S.Ct. 1855, 75 L.Ed.2d 903 (1983)(hold-ing that penal statutes must define criminal offenses with “sufficient definiteness,” and “in a manner that does not encourage arbitrary and discriminatory enforcement”); Winters v. New York, 333 U.S. 507, 515, 68 S.Ct. 665, 92 L.Ed. 840 (1948)(holding that where a statute imposes criminal.penalties, the standard of certainty involved in vagueness review is higher). In addition to defining a core of proscribed behavior to give people constructive notice of the law, a criminal statute must provide standards to prevent arbitrary enforcement. City of Chicago v. Morales, 527 U.S. 41, 52, 119 S.Ct. 1849, 144 L.Ed.2d 67 (1999). Without such standards, a statute would be impermissibly vague even if it did not reach a substantial amount of constitutionally protected conduct, because it would subject people to *1012the risk of arbitrary deprivation of their liberty. Id. Regardless of what type of conduct the criminal statute targets, the arbitrary deprivation of liberty is itself offensive to the Constitution’s due process guarantee. Smith v. Goguen, 415 U.S. 566, 575, 94 S.Ct. 1242, 39 L.Ed.2d 605 (1972).
The district court correctly applied these principles in this case. It recognized that a challenged statute enjoys a presumption of constitutionality. Baggett v. Bullitt, 377 U.S. 360, 372, 84 S.Ct. 1316, 12 L.Ed.2d 377 (1964). But where a statute criminalizes conduct, the law may not be impermissibly vague in any of its applications. Kolender, 461 U.S. at 357, 103 S.Ct. 1855, 75 L.Ed.2d 903; Jane L., 61 F.3d at 1500.
The district court concluded that these criminal statutes fail to establish any “core” of unquestionably prohibited activities. It explained this conclusion with reference to three of the statute’s key terms: “experimentation,” “investigation” and “routine,” none of which the statute defines. With respect to “experimentation,” the district court pointed out two difficulties. First, the term is ambiguous, lacking a precise definition to focus application of the statute. Forbes, 71 F.Supp.2d at 1019, citing Jane L., 61 F.3d at 1500. Second, the distinction between experimentation and treatment changes over time. Id., citing Margaret S., 794 F.2d at 999. The district court also found the term “investigation” to be ambiguous, since common definitions of the term can encompass pure research as well as more common, therapeutic medical techniques. Id. In examining the statute’s use of “routine pathological examinations” to carve out an exception to criminal liability, the district court determined that the term “routine” was also ambiguous. Id., at 1020. The statute itself does not define “routine,” see A.R.S. § 36-2302, nor does the medical community provide any official standards to help. The district court was thus concerned that any examination of post-abortion fetal tissue beyond simply mounting fetal tissue on a slide could expose doctors to criminal liability.
The district court relied upon the decisions of our sister circuits and held they applied to the contentions of the plaintiffs in this case. See Jane L., 61 F.3d at 1500 (finding that a Utah statute prohibiting “experimentation” on “live unborn chib-dr en” was void for vagueness); Margaret S, 794 F.2d at 998 (holding that a Louisiana statute prohibiting “experimentation” on unborn child or post-abortion fetal tissue also was vague to the point of being unconstitutional, in part because it did not distinguish between medical experiments and medical tests); Lifchez, 735 F.Supp. at 1363 (holding unconstitutionally vague an Illinois statute that prohibited “experimentation” on human fetuses unless such activity was “therapeutic” to the fetus).
The state in this appeal endeavors to distinguish the statutes involved in those cases on the ground that those statutes were not limited to fetal experimentation and investigation occurring after abortions. The vagueness of the words when applied to medical procedures is exactly the same, however, regardless of whether the fetus has been aborted or not.
The state also contends that the statute is clear, because a doctor can avoid violating the statute by performing no tests or other procedure on fetal tissue from induced abortions. This argument ignores the exceptions built into the statute that creates the confusion. For example, it is not clear if a doctor would run afoul of the statute if called upon to perform a DNA test involving post-abortion fetal tissue to test for paternity, or to diagnose a medical condition unrelated to the patient’s decision to have an abortion.
Under both the Arizona statute and the statutes invalidated in our sister circuits, doctors might undertake a procedure involving fetal tissue that they consider to be primarily therapeutic, perhaps even routine, but the state might consider such a procedure illegal under the statute. The distinction between experiment and treat*1013ment in the use of fetal tissue is indeterminate, regardless of whether the tissue is obtained after an induced abortion. That distinction is not clarified by the statute’s scienter requirement. See A.R.S. § 36-2302 (providing that a person shall not “knowingly use any human fetus ... for any medical experimentation ...”). A doctor might knowingly use fetal tissue from an induced abortion for a test that the physician considers primarily in furtherance of a patient’s medical interest, but which the state considers to be impermissible. Neither the statute nor the record before the district court provide any clues about how the statute would be applied to such a test.
A criminal statute such as A.R.S. § 36-2302 that prohibits medical experimentation but provides no guidance as to where the state should draw the line between experiment and treatment gives doctors no constructive notice, and gives police, prosecutors, juries, and judges no standards to focus the statute’s reach. The dearth of notice and standards for enforcement arising from the ambiguity of the words “experimentation,” “investigation,” and “routine” thus renders the statute unconstitutionally vague. Kolender, 461 U.S. at 358, 103 S.Ct. 1855, 75 L.Ed.2d 903, Papachristou v. City of Jacksonville, 405 U.S. 156, 162, 92 S.Ct. 839, 31 L.Ed.2d 110 (1972).
The state also contends that these particular statutes are not impermissibly vague, at least not in Arizona, because the Arizona physicians in this record do not harbor any uncertainty or disagreement about what procedures they will in fact avoid in light of this statute. This does not mean that the statute has any more clarity than the statutes struck down by other circuits; it does not. This means only that at this particular stage of medical research, the physicians do not disagree about the risks of prosecution they are willing to endure.
The judgment of the district court is AFFIRMED.