  United States Court of Appeals
      for the Federal Circuit
              __________________________

   SIEMENS MEDICAL SOLUTIONS USA, INC.,
           Plaintiff-Cross Appellant,
                           v.
  SAINT-GOBAIN CERAMICS & PLASTICS, INC.,
             Defendant-Appellant.
              __________________________

                   2010-1145, -1177
              __________________________

    Appeals from the United States District Court for the
District of Delaware in Case No. 07-CV-0190, Judge Sue
L. Robinson.
               __________________________

              Decided: February 24, 2011
              __________________________

    CARTER G. PHILLIPS, Sidley Austin LLP, of Washing-
ton, DC, argued for plaintiff-cross appellant. With him
on the brief were VIRGINIA A. SEITZ and JAMES C. OWENS.
Of counsel on the brief was ARTHUR I. NEUSTADT, Oblon
Spivak McClelland Maier & Neustadt, L.L.P., of Alexan-
dria, Virginia.

    GREGG F. LOCASCIO, Kirkland & Ellis, LLP, of Wash-
ington, DC, argued for defendant-appellant. With him on
the brief were JOHN C. O’QUINN, WILLIAM H. BURGESS,
SIEMENS MEDICAL   v. SAINT GOBAIN                         2


MICHAEL A. PEARSON, JR.; and ROBERT G. KRUPKA, of Los
Angeles, California.
               __________________________

    Before LOURIE, LINN, and PROST, Circuit Judges.
   Opinion for the court filed by Circuit Judge LOURIE.
    Dissenting opinion filed by Circuit Judge PROST.
LOURIE, Circuit Judge.
    Saint-Gobain Ceramics & Plastics, Inc. (“Saint-
Gobain”) appeals from the decision of the United States
District Court for the District of Delaware denying Saint-
Gobain’s motion for judgment as a matter of law (“JMOL”)
or for a new trial following the jury’s finding that Saint-
Gobain infringed U.S. Patent 4,958,080 (the “’080 patent”)
under the doctrine of equivalents. Siemens Med. Solu-
tions USA, Inc. v. Saint-Gobain Ceramics & Plastics, Inc.,
615 F. Supp. 2d 304 (D. Del. 2009) (“JMOL Opinion”).
Siemens Medical Solutions USA, Inc. (“Siemens”) cross-
appeals from the district court’s grant of Saint-Gobain’s
motion for JMOL reducing the jury’s award of damages.
Id.
    Because we discern no legal error in the court’s jury
instructions and no abuse of discretion in the court’s
evidentiary rulings, we affirm the decision of the district
court denying Saint-Gobain’s motion for JMOL or a new
trial. However, for the reasons stated below, we vacate
the district court’s damages award and remand for a
determination of additional damages.
                       BACKGROUND
    Positron emission tomography (“PET”) is a nuclear
medical imaging technique that provides images and
information about the chemical structure and function of
a patient’s organ systems. JMOL Opinion at 307. To
3                           SIEMENS MEDICAL   v. SAINT GOBAIN


obtain a PET scan, a patient is administered a radioactive
isotope and enters the PET scanner. As the isotope
decays in the patient’s body, it emits positrons. When a
positron collides with an electron in the patient’s body,
two gamma-ray photons are produced, which exit the
patient’s body. The photons are detected by the PET
scanner’s radiation detector, which contains thousands of
scintillator crystals arranged in a ring around the pa-
tient’s body. As the district court found, “[a] scintillator,
generally, is a substance that absorbs high energy radia-
tion and, in response, fluoresces photons at a specific,
longer wavelength, releasing the previously absorbed
energy.” Id. at 307 n.1. The scintillator crystals convert
the emitted gamma rays into visible light, which is then
detected to produce a three-dimensional image of the
radioactivity in the patient. This image provides useful
diagnostic information about the patient’s organ systems.
Id. at 307.
     Siemens develops, manufactures, and sells PET scan-
ners. The scintillator crystals in Siemens’ PET scanners
consist of cerium-doped lutetium oxyorthosilicate (“LSO”).
JMOL Opinion at 307. Saint-Gobain manufactures and
sells scintillator crystals for use in PET scanners. Saint-
Gobain’s crystals consist of cerium-doped lutetium-
yttrium orthosilicate (“LYSO”), which differs chemically
from LSO in that some of the lutetium is substituted for
by yttrium. Id. Specifically, Saint-Gobain’s crystals are
10% Y LYSO, meaning that 10% of the lutetium atoms
are substituted for by yttrium atoms. Saint-Gobain sells
its scintillator crystals to Philips Medical Systems (“Phil-
ips”), which manufactures and sells PET scanners that
compete with those sold by Siemens. Id. at 308.
    Siemens owns the ’080 patent, which relates to radia-
tion detectors comprising an LSO scintillator crystal and
a photodetector. JMOL Opinion at 307; J.A. 1210-11.
SIEMENS MEDICAL   v. SAINT GOBAIN                           4


The ’080 patent was filed on August 4, 1989 and expired
on October 6, 2008. JMOL Opinion at 307. Claim 1 of the
’080 patent reads as follows:
        1. A gamma ray or x-ray detector, comprising:
    a scintillator composed of a transparent single
    crystal of cerium-activated lutetium oxyorthosili-
    cate     having     the    general    formulation
    Ce2xLu2(1-x)SiO5, where x is within the range of
    from approximately 2×10–4 to approximately
    3×10–2, and
        a photodetector optically coupled to the scintil-
    lator for producing an electrical signal in response
    to the emission of a light pulse by the scintillator.
’080 patent col.12 ll.7-15.
    Although the ’080 patent is the only asserted patent
in this case, the parties’ arguments on appeal involve two
additional patents. U.S. Patent 6,624,420 (the “’420
patent”), which is licensed to Saint-Gobain, discloses
radiation detectors comprising LYSO scintillator crystals.
JMOL Opinion at 307-08. Claim 1 of the ’420 patent
recites:
        1. A scintillator detector for high energy ra-
    diation comprising: a monocrystalline structure of
    cerium doped lutetium yttrium orthosilicate,
    Ce2x,(Lu1-yYy)2(1-x)SiO5 where x=approximately
    0.0001      to       approximately    0.05    and
    y=approximately 0.0001 to approximately 0.9999.
’420 patent col.7 ll.39-43. As the formula in claim 1
indicates, the LYSO crystals disclosed in the ’420 patent
range from 0.01% Y to 99.99% Y. The specification of the
’420 patent states that LYSO crystals comprising 30% Y,
50% Y, 70% Y, and 85% Y were prepared. Id. col.4 ll.51-
58. The ’420 patent was filed on February 17, 2000, and
5                           SIEMENS MEDICAL   v. SAINT GOBAIN


its front page lists the ’080 patent among the “References
Cited.”
    The parties also reference U.S. Patent 6,323,489 (the
“’489 patent”), another patent disclosing LYSO scintillator
crystals. The ’489 patent was the subject of an interfer-
ence action with the patent application that issued as the
’420 patent. Claim 1 of the ’489 patent recites:
        1. A crystal scintillator comprising a trans-
    parent single crystal of cerium-activated lutetium
    yttrium oxyorthosilicate having the general for-
    mula Lu(2-x-z)YxCezSiO5, wherein 0.05≤x≤1.95 and
    0.001≤z≤0.02.
’489 patent col.5 ll.2-5. Unlike the ’420 patent, the speci-
fication of the ’489 patent discloses the preparation of a
crystal consisting of 10% Y LYSO. Id. col.3 l.64–col.4 l.1.
The ’489 patent was surrendered following an adverse
decision in the interference proceeding. JMOL Opinion at
307 n.3.
    In April 2007, Siemens sued Saint-Gobain for in-
fringement, including willful infringement, of claims 1
and 2 of the ’080 patent. JMOL Opinion at 307. Siemens’
infringement theory relied on the doctrine of equivalents.
Siemens alleged that Saint-Gobain was liable for con-
tributory and induced infringement under 35 U.S.C.
§§ 271(b) and (c) by selling 10% Y LYSO crystals to Phil-
ips for use in its PET scanners. Id. at 308-09. Saint-
Gobain argued in response that PET scanners comprising
its 10% Y LYSO crystals did not infringe the ’080 patent
under the doctrine of equivalents. 1 Saint-Gobain argued,
in particular, that its LYSO crystals are not equivalent to

    1    The parties did not dispute that, aside from the
scintillator crystals, the PET scanners at issue literally
satisfied all other limitations of claims 1 and 2 of the ’080
patent. JMOL Opinion at 308.
SIEMENS MEDICAL   v. SAINT GOBAIN                         6


those claimed in the ’080 patent, because its crystals are
separately claimed in the ’420 patent. Id. at 308. In
defense to Siemens’ allegation of willfulness, Saint-
Gobain pointed out that it obtained a license to the ’489
patent and then, after the interference, to the ’420 patent.
Id. at 311-12.
    A jury trial was held from September 17 to 25, 2008.
Id. Despite Saint-Gobain’s request for a higher eviden-
tiary standard to prove equivalence (an issue we discuss
further below), the court instructed the jury that Siemens
must prove contributory and induced infringement under
the doctrine of equivalents by a preponderance of the
evidence. Id. at 309. Regarding the ’420 patent, the court
instructed the jury as follows:
        [Y]ou have heard evidence that Saint-Gobain
   has a license under the [’420] patent to produce its
   10% Y LYSO crystal. In connection with this evi-
   dence, I instruct you that a product that is covered
   by a subsequent patent may still infringe an ear-
   lier patent. Nonetheless, in considering the issue
   of infringement under the doctrine of equivalents,
   you may consider that Saint-Gobain obtained the
   license under the [’420] patent, which may be
   some evidence that the differences between the
   10% Y LYSO crystal and the claimed LSO crystal
   are substantial. Such evidence may be considered
   along with other evidence of the similarities and
   differences between the claimed LSO crystal and
   Saint-Gobain’s 10% Y LYSO crystal. It is for you
   to decide the issue of whether Saint-Gobain’s 10%
   Y LYSO crystal constitutes an equivalent to the
   claimed LSO crystal of the ’080 patent.
Id.; J.A. 451. The court also instructed the jury that, in
the event it found infringement, it could consider a rea-
7                           SIEMENS MEDICAL   v. SAINT GOBAIN


sonable royalty and lost profits as damages.          JMOL
Opinion at 309; J.A. 467-70.
    The jury found in favor of Siemens on infringement
and awarded total damages of $52.3 million. 2 Id. The
jury also found that Saint-Gobain’s infringement was not
willful. Id. Saint-Gobain filed a post-trial motion for
JMOL or a new trial, in which Saint-Gobain challenged
the following decisions of the district court: (1) a decision
not to instruct the jury that infringement by equivalence
must be proved in this case by clear and convincing evi-
dence; (2) a decision not to instruct the jury that the ’420
patent is presumed valid; (3) a decision not to admit into
evidence the ’489 patent and certain expert testimony;
and (4) a decision to permit the jury to consider lost
profits damages. Saint-Gobain further argued that the
court should remit the jury’s damages award.
     In its opinion and order dated May 15, 2009, the dis-
trict court denied Saint-Gobain’s post-trial motion for
JMOL or for a new trial. JMOL Opinion at 320. How-
ever, the court granted Saint-Gobain’s request to reduce
the jury’s damages award, finding that evidence sup-
ported the jury verdict only up to $44,937,545. Id. at 319.
In particular, the court found substantial evidence that
Saint-Gobain sold 79 PET scanners’ worth of crystals to
Philips and that Philips sold 61 scanners prior to the
expiration of the ’080 patent, but that Philips’ sale of the
additional 18 scanners was “wholly speculative.” Id. at
318-19. The court therefore awarded lost profits damages
corresponding to 61 scanners based on the per scanner



    2   The verdict form did not ask the jury to specify
whether its award resulted from a calculation of reason-
able royalty, lost profits, or a combination of the two. J.A.
476-77.
SIEMENS MEDICAL   v. SAINT GOBAIN                         8


profit estimate of Siemens’ damages expert. 3 Id. at 319.
The court entered final judgment on December 16, 2009.
   The parties timely appealed and cross-appealed. We
have jurisdiction pursuant to 28 U.S.C. § 1295(a)(1).
                        DISCUSSION
    Saint-Gobain appeals the district court’s denial of its
motion for JMOL or for a new trial. In particular, Saint-
Gobain challenges the district court’s jury instructions
with regard to the doctrine of equivalents and the pre-
sumption of validity, the decision of the district court to
exclude certain evidence, and the district court’s decision
to permit the jury to consider lost profits damages. Sie-
mens cross-appeals the district court’s reduction of the
jury’s damages award.
    We review the denial of a motion for JMOL or for a
new trial under the law of the pertinent regional circuit.
Seachange Int’l, Inc. v. C-Cor Inc., 413 F.3d 1361, 1367-68
(Fed. Cir. 2005). The Third Circuit exercises plenary
review of a denial of JMOL, applying the same standard
as the district court. Curley v. Klem, 499 F.3d 199, 205-06
(3d Cir. 2007); Rinehimer v. Cemcolift, Inc., 292 F.3d 375,
383 (3d Cir. 2002). In the Third Circuit, “[t]he standard of
review on a motion for a new trial is abuse of discretion
unless the court’s denial of the motion is based on applica-
tion of a legal precept, in which case our review is ple-
nary.” Curley, 499 F.3d at 206 (internal quotation marks
omitted); Lightning Lube, Inc. v. Witco Corp., 4 F.3d 1153,
1167 (3d Cir. 1993).



   3    The court also awarded lost profits damages for
service contracts on PET scanners sold in the United
States; these damages are included in the total of
$44,937,545. JMOL Opinion at 319.
9                           SIEMENS MEDICAL   v. SAINT GOBAIN


                   A. Jury Instructions
      Whether a jury instruction is legally erroneous is a
question of law. Advanced Display Sys., Inc. v. Kent State
Univ., 212 F.3d 1272, 1282 (Fed. Cir. 2000). “This court
reviews the legal sufficiency of jury instructions on an
issue of patent law without deference to the district court.
. . . and only orders a new trial when errors in the instruc-
tions as a whole clearly mislead the jury.” DSU Med.
Corp. v. JMS Co., 471 F.3d 1293, 1304 (Fed. Cir. 2006) (en
banc in relevant part) (internal quotation marks omitted).
Moreover, a jury verdict will be set aside, based on erro-
neous jury instructions, if the movant can establish that
“those instructions were legally erroneous” and that “the
errors had prejudicial effect.” Advanced Display, 212 F.3d
at 1281.
1. Burden of Proof for Infringement under the Doctrine of
                       Equivalents
    Saint-Gobain argues that the district court legally
erred in its jury instructions regarding proof of infringe-
ment under the doctrine of equivalents. Saint-Gobain
asserts that, in cases where an alleged equivalent is
separately patented, a heightened evidentiary burden is
required. In particular, Saint-Gobain argues that, be-
cause its 10% Y LYSO crystals are covered by the ’420
patent, the district court erred by not instructing the jury
that a finding of equivalence between the 10% Y LYSO
crystals and the LSO crystals of the ’080 patent necessi-
tates proof beyond that of a preponderance of the evi-
dence.
    Saint-Gobain’s rationale for this heightened burden is
the following: The jury’s finding of equivalence “construc-
tively invalidated” the ’420 patent, because “a legal con-
clusion of invalidity for obviousness was . . . the clearly
implied result of the jury’s verdict.” Opening Br. of Def.-
SIEMENS MEDICAL   v. SAINT GOBAIN                           10


Appellant Saint-Gobain at 21. Thus, due to this implied
result, the district court legally erred by failing to instruct
the jury that equivalence must be found by clear and
convincing evidence—i.e., the evidentiary burden required
to overcome the statutory presumption of validity under
35 U.S.C. § 282. In support of its argument, Saint-Gobain
relies on Festo Corp. v. Shoketsu Kinzoku Kogyo Kabu-
shiki Co., in which we stated, “when a device that incor-
porates the purported equivalent is in fact the subject of a
separate patent, a finding of equivalency, while perhaps
not necessarily legally foreclosed, is at least considerably
more difficult to make out.” 493 F.3d 1368, 1379-80 (Fed.
Cir. 2007) (internal footnote omitted). On account of the
allegedly erroneous jury instruction, Saint-Gobain asks us
to set aside the jury verdict of infringement and order a
new trial.
     In response, Siemens argues that, even in cases of
separate patentability, infringement by equivalence is
proved by a preponderance of the evidence. According to
Siemens, our cases consistently hold that although the
fact of separate patentability may be considered by the
jury along with other relevant evidence of noninfringe-
ment, this fact does not alter the required burden of proof.
Siemens argues that the statement in Festo is dictum and
points instead to cases including Hoechst Celanese Corp v.
BP Chems. Ltd., 78 F.3d 1575, 1582 (Fed. Cir. 1996), in
which we explained that “[t]he fact of separate patentabil-
ity presents no legal or evidentiary presumption of nonin-
fringement.” Because the jury was correctly instructed,
Siemens argues, the verdict of infringement should be
affirmed.
    The doctrine of equivalents prohibits one from avoid-
ing infringement liability by making only “insubstantial
changes and substitutions . . . which, though adding
nothing, would be enough to take the copied matter
11                         SIEMENS MEDICAL   v. SAINT GOBAIN


outside the claim, and hence outside the reach of law.”
Graver Tank & Mfg. Co. v. Linde Air Prods. Co., 339 U.S.
605, 607 (1950). The “essential inquiry” in any determi-
nation under the equivalents doctrine is whether “the
accused product or process contain[s] elements identical
or equivalent to each claimed element of the patented
invention.” Warner-Jenkinson Co., Inc. v. Hilton Davis
Chem. Co., 520 U.S. 17, 40 (1997). We have assessed the
insubstantiality of an alleged equivalent by applying the
function-way-result test as set forth in Union Paper-Bag
Mach. Co. v. Murphy, 97 U.S. 120, 125 (1877), which asks
whether an element of an accused product “performs
substantially the same function in substantially the same
way to obtain the same result” as an element of the pat-
ented invention. See, e.g., TIP Sys., LLC v. Phillips &
Brooks/Gladwin, Inc., 529 F.3d 1364, 1376 (Fed. Cir.
2008); Abraxis Biosci., Inc. v. Mayne Pharma (USA) Inc.,
467 F.3d 1370, 1379 (Fed. Cir. 2006); see also Graver
Tank, 339 U.S. at 608. We remain mindful, however, that
“‘[e]quivalence, in the patent law, is not the prisoner of a
formula and is not an absolute to be considered in a
vacuum.’” Warner-Jenkinson, 520 U.S. at 24-25 (quoting
Graver Tank, 339 U.S. at 609).
    Saint-Gobain makes an interesting argument, not il-
logical, (and ably articulated by the dissent) regarding a
correspondence between the nonobviousness of an accused
product, as shown by its separate patentability, and its
infringement of another patent under the doctrine of
equivalents. However, we agree with Siemens that the
district court did not legally err by instructing the jury
that infringement in this case may be proved by a pre-
ponderance of the evidence.         Patent infringement,
whether literal or by equivalence, is an issue of fact,
which the patentee must prove by a preponderance of the
evidence. Cross Med. Prods., Inc. v. Medtronic Sofamor
SIEMENS MEDICAL   v. SAINT GOBAIN                          12


Danek, Inc., 424 F.3d 1293, 1310 (Fed. Cir. 2005) (“To
prove direct infringement, the plaintiff must establish by
a preponderance of the evidence that one or more claims
of the patent read on the accused device literally or under
the doctrine of equivalents.”); see also Morton Int’l, Inc. v.
Cardinal Chem. Co., 5 F.3d 1464, 1468 (Fed. Cir. 1993);
SRI Int’l v. Matsushita Elec. Corp. of Am., 775 F.2d 1107,
1123 (Fed. Cir. 1985) (en banc) (“The patentee bears the
burden of proving infringement by a preponderance of the
evidence.”). Saint-Gobain nonetheless insists that in one
particular situation—where the purported equivalent is
claimed in a separately issued United States patent—
proof of infringement by equivalence requires clear and
convincing evidence. As we shall explain, however, we
decline to inject Saint-Gobain’s proposed rule into the
interstices of our longstanding equivalence doctrine
jurisprudence.
    Saint-Gobain argues that Festo mandates an in-
creased burden of proof in cases of separate patentability,
but Festo did not involve this issue. In Festo, in the
context of prosecution history estoppel as a legal limita-
tion on the doctrine of equivalents, our sole consideration
was whether an equivalent was unforeseeable as of the
patent application’s filing date. Festo, 493 F.3d at 1377.
Nonetheless, Saint-Gobain points out that, in rejecting
one party’s proposed legal standard for foreseeability, we
stated:
    We have not directly decided whether a device—
    novel and separately patentable because of the in-
    corporation of an equivalent feature—may be cap-
    tured by the doctrine of equivalents, although we
    have held that when a device that incorporates
    the purported equivalent is in fact the subject of a
    separate patent, a finding of equivalency, while
    perhaps not necessarily legally foreclosed, is at
13                         SIEMENS MEDICAL   v. SAINT GOBAIN


     least considerably more difficult to make out. But
     there is a strong argument that an equivalent
     cannot be both non-obvious and insubstantial.
Id. at 1379-80 (internal footnotes omitted). Saint-Gobain
also relies upon the concurring opinion in Roton Barrier,
Inc. v. Stanley Works, in which Judge Nies wrote that “a
second patent, depending on its subject matter, may be
relevant to the issue of whether the changes [in an ac-
cused device] are substantial,” and that “[a] substitution
in a patented invention cannot be both nonobvious and
insubstantial.” 79 F.3d 1112, 1128 (Fed. Cir. 1996) (Nies,
J., additional views).
     These passages, however, cannot reasonably be read
to require proof of equivalency by clear and convincing
evidence in cases of separate patentability. Rather, these
statements indicate that where, as here, the alleged
equivalent is claimed in a separate patent, this fact, when
weighed by the fact-finder together with all other relevant
evidence, may make equivalency “considerably more
difficult to make out” by a preponderance of the evidence.
     The issue of infringement by a separately patented
equivalent was addressed in the Supreme Court decision
of Sanitary Refrigerator Co. v. Winters, 280 U.S. 30
(1929). In that case, the patent in suit, filed by Winters
and Crampton, claimed an improved refrigerator door
latch that automatically locked as the door closed. Id. at
36-37. The allegedly infringing door latch was covered by
a later-issued U.S. patent to Schrader. In its opinion, the
Court first acknowledged that the accused device did not
literally infringe the narrow claims of the Winters and
Crampton patent. Id. at 41. The Court then proceeded to
apply the function-way-result test to the accused device
and found infringement by equivalence. Id. at 41-42. The
Court held that infringement cannot be avoided in a
SIEMENS MEDICAL   v. SAINT GOBAIN                         14


device that has “no substantial departure” from the
claimed invention. Id. at 42. The Court further noted:
“Nor is the infringement avoided . . . by any presumptive
validity that may attach to the Schrader patent by reason
of its issuance after the Winters and Crampton patent.”
Id. at 43. The Court thus found equivalence in the face of
separate patentability without imposing a heightened
evidentiary burden.
    The passage in Festo relied on by Saint-Gobain in-
cludes citations of numerous cases in which we, like the
Court in Sanitary Refrigerator, rejected arguments that
separate patentability warrants anything more than
consideration of this fact together with all others weigh-
ing for and against equivalency. Festo, 493 F.3d at 1379
n.8 (citing Glaxo Wellcome, Inc. v. Andrx Pharms., Inc.,
344 F.3d 1226 (Fed. Cir. 2003); Hoechst Celanese, 78 F.3d
1575; Nat’l Presto Indus. v. W. Bend Co., 76 F.3d 1185
(Fed. Cir. 1996); Atlas Powder Co. v. E.I. du Pont De
Nemours & Co., 750 F.2d 1569 (Fed. Cir. 1984)).
    In Hoechst Celanese, for example, the accused in-
fringer argued that the fact that it “is practicing a process
that is separately patentable . . . is presumptive evidence
of non-infringement,” both literally and by equivalence.
78 F.3d at 1582. We disagreed, noting that “[t]he fact of
separate patentability presents no legal or evidentiary
presumption of noninfringement.” Id. In Glaxo Wellcome,
we vacated a district court’s grant of summary judgment
of noninfringement, both literally and under the doctrine
of equivalents. 344 F.3d at 1233. In doing so, we rejected
the accused infringer’s argument that it did not infringe
because its accused pharmaceutical formulation was
separately patented. We explained that, “[a]lthough this
fact may be weighed by the district court, particularly if
there is an issue of ‘insubstantial’ change with respect to
15                          SIEMENS MEDICAL   v. SAINT GOBAIN


equivalency, separate patentability does not automati-
cally negate infringement.” Id.
    Moreover, in National Presto, we confirmed that “[t]he
fact of separate patentability is relevant, and is entitled to
due weight,” but does not confer any presumption of
noninfringement; rather, “[s]uch evidence when present
warrants consideration by the trier of fact, along with the
other evidence of the differences and similarities of the
patented and accused devices.” 76 F.3d at 1192. In Atlas
Powder, we rejected the argument that the grant of a
patent to an accused infringer constitutes a prima facie
determination of nonequivalence. 750 F.2d at 1580.
Instead, we endorsed the view that “‘[p]atentable differ-
ence does not of itself tend to negative infringement.’” Id.
at 1581 (quoting Herman v. Youngstown Car Mfg. Co.,
191 F. 579, 585 (6th Cir. 1911)).
     None of these cases cited in Festo supports a require-
ment of proof of equivalency by clear and convincing
evidence in cases involving separate patentability. In-
deed, in other cases we have likewise indicated that
separate patentability, while potentially relevant to the
equivalence issue and deserving of due weight in the
infringement analysis, does not merit a heightened evi-
dentiary burden. See, e.g., Abraxis Biosci., 467 F.3d 1370,
1382 (Fed. Cir. 2006) (affirming the district court’s finding
of infringement by equivalence and stating that separate
patentability of the accused pharmaceutical formulation
did not outweigh substantial evidence of its equivalence);
Fiskars, Inc. v. Hunt Mfg. Co., 221 F.3d at 1324 (holding
that the trial court judge did not err by withholding
evidence of separate patentability from the jury, because
“it is well established that separate patentability does not
avoid equivalency as a matter of law”); Zygo Corp. v. Wyko
Corp., 79 F.3d 1563, 1570 (Fed. Cir. 1996) (“The nonobvi-
ousness of the accused device, evidenced by the grant of a
SIEMENS MEDICAL   v. SAINT GOBAIN                        16


United States patent, is relevant to the issue of whether
the change therein is substantial.”).
    Saint-Gobain argues that proof by clear and convinc-
ing evidence is required on the facts of this case, because
the jury’s finding of equivalence “constructively invali-
dates” the ’420 patent. We disagree with Saint-Gobain’s
reasoning in several respects. First, patents are afforded
a statutory presumption of validity under 35 U.S.C. § 282,
and overcoming this presumption requires proof by clear
and convincing evidence. See, e.g., Pfizer, Inc. v. Apotex,
Inc., 480 F.3d 1348, 1359-60 (Fed. Cir. 2007); Hybritech
Inc. v. Monoclonal Antibodies, Inc., 802 F.2d 1367, 1375
(Fed. Cir. 1986). Even if equivalence were relevant to
obviousness, a point on which we need not and do not
express an opinion, the jury properly found infringement
under the equivalence doctrine by only a preponderance of
the evidence. Thus, ipso facto, the jury’s finding could not
invalidate the ’420 patent, constructively or otherwise.
    Moreover, with regard to Saint-Gobain’s contention
that equivalence is tantamount to obviousness, we dis-
agree. The two legal principles require different analyti-
cal frameworks. The doctrine of equivalents, although
“not the prisoner of a formula,” Graver Tank, 339 U.S. at
609, typically involves application of the insubstantial
differences test, usually via the function-way-result test.
Obviousness, by contrast, requires analysis under the four
Graham factors. Graham v. John Deere Co., 383 U.S. 1,
17-18 (1966); accord KSR Int’l Co. v. Teleflex Inc., 550
U.S. 398, 406-07 (2007). Unlike equivalence, an obvious-
ness inquiry may center on, for example, objective evi-
dence of commercial success, e.g., Symbol Techs., Inc. v.
Opticon, Inc., 935 F.2d 1569, 1579 (Fed. Cir. 1991), or on
the level of predictability in the art, e.g., Sanofi-
Synthelabo v. Apotex, Inc., 550 F.3d 1075, 1089 (Fed. Cir.
2008). Indeed, although we express no view regarding the
17                         SIEMENS MEDICAL   v. SAINT GOBAIN


validity of the ’420 patent, we note that Saint-Gobain’s
“constructive invalidity” argument fails to account for the
commercial success of its own 10% Y LYSO crystals or the
unpredictability in the field of crystallization. See, e.g.,
id. (noting experts’ agreement regarding the unpredict-
ability of crystallization).
    Furthermore, the time frames of the two inquiries dif-
fer. Under the doctrine of equivalents, “the proper time
for evaluating equivalency . . . is at the time of infringe-
ment, not at the time the patent was issued,” Warner-
Jenkinson, 520 U.S. at 37, yet obviousness asks whether a
claimed invention “would have been obvious at the time
the invention was made,” 35 U.S.C. § 103(a). These and
other differences between equivalence and obviousness
undermine Saint-Gobain’s theory of “constructive invali-
dation.”
    Saint-Gobain’s professed justification for a clear and
convincing standard also overlooks an important distinc-
tion regarding claim scope. The jury’s infringement
verdict indicates only that Saint-Gobain’s marketed 10%
Y LYSO crystals are equivalent to the LSO crystals of
claim 1 or 2 of the ’080 patent. JMOL Opinion at 308-09.
Importantly, the verdict does not imply equivalence
between the ’080 patent and the full claim scope of the
’420 patent—which, as noted above, covers LYSO crystals
ranging from 0.01% Y to 99.99% Y. An invalidity analysis
under § 103 requires a comparison between the prior art
and the claimed “subject matter as a whole, not separate
pieces of the claim.” Sanofi-Synthelabo, 550 F.3d at 1086.
The jury made no such comparison under the doctrine of
equivalents, further highlighting the fallacy of Saint-
Gobain’s premise that the ’420 patent was “constructively
invalidated.”
SIEMENS MEDICAL   v. SAINT GOBAIN                          18


    In summary, we agree with neither the proposed rule
nor the rationale proffered by Saint-Gobain. We, like the
district court, “decline[] to be the first (and only) court to
depart from an extended history of patent infringement
jurisprudence applying the preponderance of the evidence
standard.” JMOL Opinion at 310. We therefore hold
that, even though the alleged equivalent in this case was
separately patented, the district court did not legally err
by instructing the jury that infringement under the
doctrine of equivalents may be proved by a preponderance
of the evidence. 4
        2. Presumption of Validity of the ’420 Patent
    Saint-Gobain also argues that, because the validity of
the ’420 patent was at issue in this case, the district court
erroneously denied its proposed jury instruction that the
’420 patent was entitled to a presumption of validity.
Saint-Gobain reasons that because the ’420 patent issued
after the patent examiner considered the ’080 patent as
prior art, 10% Y LYSO crystals are necessarily nonobvi-
ous and patentable over LSO crystals. Saint-Gobain
equates the issuance of the ’420 patent to a determination
by the Patent and Trademark Office (“PTO”) that 10% Y
LYSO and LSO crystals are not substantially equivalent.
Thus, Saint-Gobain argues, instructing the jury that the
’420 patent is presumed valid “would have made proof of
equivalence more difficult, as Festo requires.” Opening
Br. of Def.-Appellant Saint-Gobain at 35. Saint-Gobain
further argues that at trial Siemens challenged the valid-
ity and enforceability of the ’420 patent, thus heightening
the need for its proposed jury instruction.

    4   Because Saint-Gobain does not appeal the jury’s
finding by a preponderance of the evidence that Saint-
Gobain infringed the ’080 patent under the doctrine of
equivalents, we do not address that issue in our opinion.
19                         SIEMENS MEDICAL   v. SAINT GOBAIN


    Siemens responds that the validity of the ’420 patent
was not at issue in this case and therefore that the pre-
sumption of validity was not applicable to the case.
Siemens also stresses that the ’420 patent was admitted
into evidence, that the jury was informed during the trial
that all patents are presumed valid, and that the district
court permitted Saint-Gobain to reiterate this point
during its closing arguments. Siemens argues that, in
connection with Saint-Gobain’s defense to willfulness and
the intent prong to inducement, Siemens was justified in
probing the circumstances under which Saint-Gobain
itself believed the ’420 patent might be invalid. Accord-
ingly, Siemens asserts that the district court properly
exercised its discretion by declining to instruct the jury
regarding the presumption of validity of the ’420 patent.
    We conclude that the district court’s decision not to
provide separate instructions to the jury about the pre-
sumption of validity did not “clearly mislead the jury.”
DSU Med., 471 F.3d at 1304 (internal quotation marks
omitted). Patent validity was not an issue before the jury:
the validity of the ’080 patent was not challenged at trial,
and, as we explained above, the verdict of infringement by
equivalence in no way affected the validity of the ’420
patent. As the district court correctly noted, Siemens
probed aspects of the ’420 patent in connection with
Saint-Gobain’s license defense to willfulness, but “appro-
priately stopped short of directly challenging the validity
of the ’420 patent.” JMOL Opinion at 313. Thus, the
requested instruction was not directly relevant to the
issues before the jury.
    To the extent that the presumed validity of the ’420
patent made a finding of infringement by equivalence
more difficult, as Saint-Gobain argues, the jury had notice
of the presumption. During the trial, the jury viewed a
video providing an overview of the patent system, which
SIEMENS MEDICAL   v. SAINT GOBAIN                          20


stated in part that “[t]o prove that a patent is invalid, the
law requires a higher standard of proof, since the PTO is
presumed to have done its job correctly.” Id.; J.A. 652. In
its closing arguments, Saint-Gobain affirmatively stated
at least three times on the record that the ’420 patent is
presumed valid. JMOL Opinion at 312; J.A. 1763-88. In
its final instructions, the district court invited the jury as
part of its equivalence analysis to “consider that [Saint-
Gobain] obtained the license under the [’420] patent,
which may be some evidence that the differences between
the 10% Y LYSO crystal[s] and the claimed LSO crystal[s]
are substantial.” JMOL Opinion at 309. We therefore
agree with the district court that the jury was not clearly
misled by the court’s ruling excluding a specific jury
instruction on the validity of the ’420 patent.
                  B. Exclusion of Evidence
     We review a district court’s decision to exclude evi-
dence under the law of the regional circuit. Del. Valley
Floral Grp., Inc. v. Shaw Rose Nets, LLC, 597 F.3d 1374,
1379 (Fed. Cir. 2010). The Third Circuit reviews a dis-
trict court’s decision to exclude evidence for abuse of
discretion. Glass v. Phila. Elec. Co., 34 F.3d 188, 191 (3d
Cir. 1994); see also Rhodia Chimie v. PPG Indus., Inc.,
402 F.3d 1371, 1376-77 (Fed. Cir. 2005). “A ruling exclud-
ing evidence under Rule 403 is accorded particular defer-
ence, and . . . it may not be reversed unless the
determination is arbitrary and irrational.” In re Paoli
R.R. Yard PCB Litig., 113 F.3d 444, 453 (3d Cir. 1997)
(internal quotation marks omitted). An erroneous eviden-
tiary ruling “is harmless only if it is highly probable that
the error did not affect the outcome of the case.” Glass, 34
F.3d at 191 (internal quotation marks omitted).
21                          SIEMENS MEDICAL   v. SAINT GOBAIN


                    1. The ’489 Patent
    Saint-Gobain argues that the district court erred by
excluding the ’489 patent from evidence. Saint-Gobain
contends that the ’489 patent demonstrates the pat-
entability, and therefore the nonequivalence, of 10% Y
LYSO. Saint-Gobain further contends that the ’489
patent served to rebut Siemens’ attack on the validity of
the ’420 patent.
    In response, Siemens argues that, because Saint-
Gobain failed to raise its relevance argument regarding
the ’489 patent at the district court, this argument is now
waived. Siemens also contends that the district court
properly excluded the ’489 patent, because its potential to
confuse the jury outweighed its potential relevance.
Finally, Siemens argues that any error associated with
the court’s ruling is harmless, because the jury was in-
formed of the existence and content of the ’489 patent.
    We agree with Siemens that the district court did not
abuse its discretion by excluding the ’489 patent from
evidence. At trial, Saint-Gobain sought to introduce the
’489 patent only in connection with its license defense to
Siemens’ claim of willful infringement. JMOL Opinion at
311-12. As the district court noted, Saint-Gobain “did not
offer the ’489 patent for a disclosure of LYSO crystals.”
Id. at 312. In response to Saint-Gobain’s request, the
district court admitted testimony regarding Saint-
Gobain’s license of the ’489 patent and the license itself,
but excluded the ’489 patent under Rule 403, reasoning
that “there is a high likelihood of confusion to the jury” in
admitting the abandoned patent. Id.
    Only after trial did Saint-Gobain argue for the first
time that the ’489 patent should have been admitted for
its relevance to the patentability of LYSO crystals. Id.
Because at the time of its evidentiary ruling the district
SIEMENS MEDICAL   v. SAINT GOBAIN                        22


court was not informed of Saint-Gobain’s view that the
’489 patent was relevant to its equivalence argument,
Saint-Gobain cannot now attack the court’s evidentiary
ruling on this basis. See Old Chief v. United States, 519
U.S. 172, 182 n.6 (1997) (“It is important that a reviewing
court evaluate the trial court's decision from its perspec-
tive when it had to rule and not indulge in review by
hindsight.”). Thus, the district court’s ruling under Rule
403 was not arbitrary and irrational.
    We also agree with Siemens that any error associated
with the district court’s ruling was harmless. The ’489
patent, which was abandoned following the interference
with the ’420 patent, was not (and could not have been)
asserted in this case. Even if Saint-Gobain had presented
its relevance arguments prior to the court’s evidentiary
ruling, the ’489 patent would largely have been cumula-
tive of evidence already of record, including the ’420
patent. Id. at 312. Both the ’420 and ’489 patents dis-
close and claim substantially the same invention—hence
the interference proceeding. Id. at 308 n.3. During the
trial, the jury was informed of the existence and content
of the ’489 patent, including Saint-Gobain’s efforts to
obtain a license (id. at 312; J.A. 940; J.A. 1244-45; J.A.
1261-62), and Saint-Gobain presented to the jury a time-
line that contained both a text reference to the ’489 pat-
ent’s issuance and an image of the patent itself (J.A.
2032). For these reasons we conclude that it is “highly
probable” that any error associated with the exclusion of
the ’489 patent did not affect the outcome of this case, and
therefore that any error in the district court’s evidentiary
ruling was harmless. Glass, 34 F.3d at 191.
                   2. Expert Testimony
    Saint-Gobain argues that the district court abused its
discretion by granting Siemens’ motion to exclude certain
23                         SIEMENS MEDICAL   v. SAINT GOBAIN


testimony of Saint-Gobain’s expert, Dr. McClellan, re-
garding the use of LYSO in applications other than PET.
Saint-Gobain argues that the ’080 patent claims a scintil-
lator without regard to its particular application, and
therefore that it was inappropriate to limit the substan-
tial equivalence inquiry solely to PET scan applications.
Saint-Gobain also points out that the importance of
different scintillation properties can vary dramatically by
application, so the district court could not properly deter-
mine substantial equivalence by examining only a single
application.
    Siemens responds that the district court did not re-
strict Dr. McClellan from testifying on the scintillation
properties Saint-Gobain now complains were excluded;
the court only prevented Dr. McClellan from going beyond
his expert disclosures or discussing matters not disclosed
to Siemens during discovery. Siemens asserts that the
court’s ruling imposed sensible limitations on expert
testimony under the Federal Rules of Civil Procedure.
Siemens contends that Dr. McClellan in fact testified at
length about many of the scintillation properties that
Saint-Gobain claims were excluded. Siemens further
contends that Saint-Gobain waived the arguments it now
makes on appeal by failing to make a specific proffer of
the testimony that Dr. McClellan would have presented at
trial.
    The district court did not abuse its discretion in its
ruling regarding the expert testimony of Dr. McClellan.
We agree that the district court “did not render a specific
ruling regarding the substance of Dr. McClellan’s testi-
mony,” JMOL Opinion at 313, but rather imposed sensi-
ble limitations on proposed testimony based upon
undisclosed data and information. In its evidentiary
ruling, the district court noted that Dr. McClellan’s ex-
perience with LSO and LYSO resulted from studies he
SIEMENS MEDICAL   v. SAINT GOBAIN                          24


performed as a government contractor at the Los Alamos
National Laboratory (“LANL”). Siemens Med. Solutions
USA, Inc. v. Saint-Gobain Ceramics & Plastics, Inc., 2008
WL 3862091, at *1 (D. Del. Aug. 20, 2008) (“Evidence
Order”). Because his research involved national security
matters, Dr. McClellan was unable to use any work-
related materials in the litigation or to produce any LANL
documents during discovery. Id. The court expressed
concern that, without any ability for Siemens to examine
the studies that formed the basis of Dr. McClellan’s
opinions, “there is clearly no principled way to test his
recollection and opinion.” Id.
    The district court further noted in its evidentiary rul-
ing that, although there were documents at LANL relat-
ing to the studies referenced in Dr. McClellan’s expert
report, he did not review these materials to refresh his
recollection when preparing his report. Id. The court
thus identified a “serious question of whether his recollec-
tion of data generated several years ago can possibly
constitute a sufficient basis for an expert opinion” under
Rule 702 of the Federal Rules of Evidence. Id. For these
reasons, the court granted Siemens’ motion to exclude
portions of Dr. McClellan’s testimony that were not dis-
closed in discovery—i.e., it ruled that Dr. McClellan could
not testify on matters not disclosed in his expert report or
deposition, and he could not rely on testing that was not
disclosed to Siemens during discovery. Id.; JMOL Opin-
ion at 314.
    The court’s evidentiary ruling was justified and well
reasoned. The court’s judgment comports with Fed. R.
Civ. P. 26(a)(2)(B)(i)-(ii), which requires experts to provide
a written report containing “a complete statement of all
opinions the witness will express and the basis and rea-
sons for them” and “the facts or data considered by the
witness in forming them.” As the Advisory Committee
25                         SIEMENS MEDICAL   v. SAINT GOBAIN


Note accompanying the 1993 amendments to Rule 26
explains: “Given this obligation of disclosure, litigants
should no longer be able to argue that materials furnished
to their experts to be used in forming their opinions—
whether or not ultimately relied upon by the expert—are
privileged or otherwise protected from disclosure when
such persons are testifying or being deposed.” Fed. R.
Civ. P. 26 advisory committee’s note.
    Rule 26, therefore, “proceeds on the assumption that
fundamental fairness requires disclosure of all informa-
tion supplied to a testifying expert in connection with his
testimony.” In re Pioneer Hi-Bred Int’l, Inc., 238 F.3d
1370, 1375 (Fed. Cir. 2001). We agree with the district
court that “the fundamental principle of fairness” sup-
ports its sensible limitations on Dr. McClellan’s testi-
mony, as Siemens had “no principled way to test his
recollection and opinion.” Evidence Order at *1.
     The court’s ruling further complies with Fed. R. Civ.
P. 37(c)(1), which states that, if a party fails to comply
with Rule 26(a), “the party is not allowed to use that
information or witness to supply evidence on a motion, at
a hearing, or at a trial, unless the failure was substan-
tially justified or is harmless.” Saint-Gobain does not
argue that its failure to disclose was substantially justi-
fied or harmless. Thus, the district court did not abuse its
discretion by ruling that Dr. McClellan could not testify
on matters not disclosed in his expert report or deposition,
and that he could not rely on testing that was not dis-
closed to Siemens during discovery.
                 C. Lost Profits Damages
    Whether lost profits are legally compensable in a par-
ticular situation is a question of law that we review de
novo. Poly-Am., L.P. v. GSE Lining Tech., Inc., 383 F.3d
1303, 1311 (Fed. Cir. 2004) (citing Rite-Hite Corp. v.
SIEMENS MEDICAL   v. SAINT GOBAIN                       26


Kelley Co., 56 F.3d 1538, 1544 (Fed. Cir. 1995) (en banc)).
“To recover lost profits, the patent owner must show
causation in fact, establishing that but for the infringe-
ment, he would have made additional profits.” Wechsler
v. Macke Int’l Trade, Inc., 486 F.3d 1286, 1293 (Fed. Cir.
2007) (internal quotation marks omitted). Thus, in gen-
eral, “the patent owner must prove (1) a demand for the
patented product, (2) an absence of acceptable noninfring-
ing substitutes, (3) the manufacturing and marketing
capability to exploit the demand, and (4) the amount of
profit the patent owner would have made.” Cohesive
Techs., Inc. v. Waters Corp., 543 F.3d 1351, 1373 (Fed.
Cir. 2008) (quoting Standard Havens Prods., Inc. v.
Gencor Indus., Inc., 953 F.2d 1360, 1373 (Fed. Cir. 1991));
see also Panduit Corp. v. Stahlin Bros. Fibre Works, Inc.,
575 F.2d 1152, 1156 (6th Cir. 1978).
     Saint-Gobain argues that the court erred by allowing
the jury to consider lost profits damages, because Siemens
failed to demonstrate that it would have made Saint-
Gobain’s sales “but for” the infringement. Saint-Gobain
does not dispute that Philips’ LYSO scanner competed
with Siemens’ LSO scanner. However, Saint-Gobain cites
evidence that General Electric (“GE”) sold scanners
containing bismuth germinate (“BGO”) scintillator crys-
tals, JMOL Opinion at 315, and argues that the market
thus consisted of three competing sellers: Siemens,
Philips, and GE. Saint-Gobain contends that GE’s BGO
scanners competed with Siemens’ LSO scanners, because
Siemens lost PET scanner sales to GE. Saint-Gobain also
argues that it could have made and sold scanners contain-
ing lanthanum bromide (LaBr3) scintillator crystals,
which were an available, noninfringing substitute for LSO
crystals.
   Siemens argues in response that sufficient evidence
supported submitting the question of lost profits to the
27                         SIEMENS MEDICAL   v. SAINT GOBAIN


jury and the jury’s award. Siemens argues that, but for
the infringement, it would have made the sales made by
Saint-Gobain. Siemens further argues that substantial
evidence supports a high-end, two-supplier PET market
consisting of Siemens’ LSO scanners and Saint-Gobain’s
LYSO scanners. Siemens contends that GE’s BGO scan-
ners were cheaper and generally inferior to LSO and
LYSO, and therefore were not competitive. In addition,
Siemens argues that it is irrelevant whether it lost sales
to GE, because the issue is what Philips’ customers would
have done had Saint-Gobain’s infringement not occurred.
Lastly, Siemens argues that the jury had ample support-
ing evidence to believe Siemens’ claim that LaBr3 was not
available and was an unacceptable alternative to LSO
and 10% Y LYSO.
    We perceive no legal error in the district court’s deci-
sion to permit the jury to award lost profits damages. On
appeal, the dispute between Saint-Gobain and Siemens
over lost profits centers on (1) whether there existed a
two-supplier, high-end market for PET scanners, and (2)
whether LaBr3 scanners were available, acceptable nonin-
fringing alternatives to LSO scanners. On both issues,
Siemens presented substantial evidence to support a lost
profits award.
    Accurately identifying a two-supplier market “‘re-
quires an analysis which excludes alternatives to the
patented product with disparately different prices or
significantly different characteristics.’” Micro Chem., Inc.
v. Lextron, Inc., 318 F.3d 1119, 1124 (Fed. Cir. 2003)
(quoting Crystal Semiconductor Corp. v. TriTech Micro-
elecs. Int’l, Inc., 246 F.3d 1336, 1356 (Fed. Cir. 2001)).
Here, the parties disputed whether GE’s BGO scanner
was an alternative to the LSO and LYSO scanners such
that a three-supplier market existed. The district court
summarized the ample evidence that BGO was generally
SIEMENS MEDICAL   v. SAINT GOBAIN                        28


inferior to LSO with respect to important scintillator
properties, including light output and decay time. JMOL
Opinion at 315-16. In addition, the district court credited
the testimony of Siemens’ head of marketing and sales,
who testified that BGO-based scanners do not compete
with Siemens’ LSO-based scanners in the high-end PET
scanner market, id. at 316, because BGO scanners, with
their relatively poor image quality, were purchased by
“customers that are extremely tight in their budgets,”
whereas the more expensive LSO scanners were pur-
chased by customers seeking the “highest performance
and high technology.” J.A. 985.
    The district court also credited Saint-Gobain’s own
documents and the testimony of Saint-Gobain’s witnesses,
which consistently described a two-supplier “high-end
PET scanner market” consisting only of LSO and LYSO
scanners. JMOL Opinion at 316. Saint-Gobain is correct
in arguing that its damages expert provided testimony
indicating that Siemens lost some sales to GE’s BGO
scanners during the relevant time period. However, as
the district court correctly noted, there was ample evi-
dence for a reasonable jury to infer the existence of a two-
supplier market for high-end PET scanners consisting of
Siemens’ LSO scanners and Philips’ LYSO scanners. Id.
    The parties also dispute whether LaBr3-based scan-
ners were available substitutes. To be “available,” an
acceptable noninfringing substitute must have been
“available or on the market” at the time of infringement.
Grain Processing Corp. v. Am. Maize-Prods. Co., 185 F.3d
1341, 1349 (Fed. Cir. 1999) (emphasis and internal quota-
tion marks omitted). A substitute need not be on sale at
the time of infringement, but if the substitute cannot be
commercialized “readily,” then it is not available for
purposes of a lost profits determination. See Micro
Chem., 318 F.3d at 1123; cf. Grain Processing, 185 F.3d at
29                         SIEMENS MEDICAL   v. SAINT GOBAIN


1354. At trial, the parties disputed whether LaBr3 was an
available alternative to LSO, and on appeal Saint-Gobain
points to evidence that it asserts supports the availability
of LaBr3. Notwithstanding, the evidence reasonably
supported a finding that LaBr3 was not an available
alternative. As the district court noted, evidence showed
that LaBr3 was at least a year and a half behind LYSO in
development. JMOL Opinion at 317. In addition, Saint-
Gobain stipulated that there were no commercial PET
scanners sold on the market using a LaBr3 scintillation
crystal. J.A. 1111. The court also cited evidence demon-
strating several disadvantages inherent to LaBr3, includ-
ing its inferior density and hygroscopicity.         JMOL
Opinion at 317.
    Because of the substantial evidence in the record sup-
porting the elements required for an award of lost profits,
we find no legal error in the district court’s decision to
permit the jury to consider lost profits damages.
 D. Cross-Appeal of the Reduction of the Jury’s Damage
                        Award
    Siemens cross-appeals the district court’s decision to
reduce the jury’s damages award. We review the decision
of a district court to reduce a jury’s damages award under
the law of the regional circuit. Minks v. Polaris Indus.,
Inc., 546 F.3d 1364, 1369-70 (Fed. Cir. 2008); Tronzo v.
Biomet, Inc., 236 F.3d 1342, 1346 (Fed. Cir. 2001). The
Third Circuit reviews a discretionary reduction in a jury
award for abuse of discretion. Cortez v. Trans Union,
LLC, 617 F.3d 688, 716 (3d Cir. 2010); see also Spence v.
Bd. of Educ. of the Christina Sch. Dist., 806 F.2d 1198,
1201 (3d Cir. 1986) (“[A trial judge’s determination] that a
decision of the jury is clearly unsupported and/or exces-
sive. . . . falls within the discretion of the trial judge,
whose decision cannot be disturbed by this court absent a
SIEMENS MEDICAL   v. SAINT GOBAIN                         30


manifest abuse of discretion.”). “The trial judge is in the
best position to evaluate the evidence and assess whether
the jury’s verdict is rationally based.” Rivera v. V.I.
Housing Auth., 854 F.2d 24, 27 (3d Cir. 1988). Accord-
ingly, the scope of review “is exceedingly narrow.” Cortez,
617 F.3d at 719.
     Siemens argues that substantial evidence supports
the jury’s full award of damages. Specifically, Siemens
contends that there was sufficient circumstantial evidence
that Philips either had arranged to sell or would in fact
sell all 79 scanners following their manufacture. Siemens
argues in the alternative that the court erred by not
awarding at least reasonable royalty damages for the
remaining 18 scanners, because evidence showed that
these 18 scanners were at least made, if not sold. Sie-
mens thus argues that the district court erred by failing
to consider that, under 35 U.S.C. § 271, it is an act of
infringement to make, use, or offer to sell a patented
invention, even if the product is not actually sold.
    Saint-Gobain responds that the court correctly re-
duced the jury’s damages award, because, as the district
court determined, the evidence is “wholly speculative”
regarding the sale of the additional 18 scanners. In
addition, Saint-Gobain argues that Siemens did not assert
a right to a reasonable royalty before the district court, so
this argument is now waived on appeal. Finally, Saint-
Gobain argues that the evidence does not support a
finding that the 18 scanners’ worth of crystals were
actually used to make scanners.
    We conclude that the district court did not abuse its
discretion in reducing the jury’s damages award regard-
ing lost profits on the additional 18 scanners. As dis-
cussed above, to recover lost profits damages, “the
patentee must show a reasonable probability that, ‘but
31                         SIEMENS MEDICAL   v. SAINT GOBAIN


for’ the infringement, it would have made the sales that
were made by the infringer.” Rite-Hite, 56 F.3d at 1545.
Here, the parties do not dispute that Saint-Gobain deliv-
ered 79 scanners’ worth of crystals to Philips, and that
Philips made and sold at least 61 scanners. JMOL Opin-
ion at 318. Rather, the dispute on appeal centers on
whether Philips actually manufactured and sold the
additional 18 scanners.
    As Siemens itself acknowledges, the only “evidence” of
infringing sales of the 18 scanners was (1) Saint-Gobain’s
delivery of 79 scanners’ worth of crystals to Philips, and
(2) the testimony of Philips’ designated corporate repre-
sentative, who agreed that “it’s not in Philips’ interest to
hold a lot of inventory of crystals” and that “[Philips
doesn’t] want to buy all of [its] supplies way in advance of
when [it] need[s] them.” JMOL Opinion at 318; J.A. 1004.
As the district court correctly found, this evidence indi-
cates that Philips generally did not stockpile crystals and
thus had incorporated them into infringing scanners—i.e.,
that Philips made the additional 18 scanners—but the
evidence does not suggest that Philips sold the additional
18 scanners prior to the expiration of the ’080 patent.
JMOL Opinion at 319. Thus, with regard to the addi-
tional 18 scanners, there is no proof of “sales that were
made by the infringer.” Rite-Hite, 56 F.3d at 1545. We
therefore agree with Saint-Gobain that the district court
did not abuse its discretion by reducing the jury’s dam-
ages award to account for the lack of substantial evidence
supporting lost profits damages associated with the
additional 18 scanners.
    However, we agree with Siemens that the court erred
by failing to consider any damages on the additional 18
scanners. A district court abuses its discretion when it
commits legal error in its award of damages. State In-
dus., Inc. v. Mor-Flo Indus., Inc., 883 F.2d 1573, 1577
SIEMENS MEDICAL   v. SAINT GOBAIN                         32


(Fed. Cir. 1989). Under 35 U.S.C. § 271, a party infringes
a patent by contributing to or inducing another party to
make, use, offer to sell, or sell a patented invention in the
United States. In addition, under 35 U.S.C. § 284, “the
floor for a damage award is no less than a reasonable
royalty . . . and the award may be split between lost
profits as actual damages to the extent they are proven
and a reasonable royalty for the remainder.” Id.; accord
Rite-Hite, 56 F.3d at 1545.
     Here, the jury determined that Saint-Gobain in-
fringed the ’080 patent by inducing and contributing to
Philips’ infringement. JMOL Opinion at 309. The dis-
trict court correctly noted that substantial evidence
supported the conclusion that Philips made the additional
18 scanners. Id. at 319. One who “makes” a patented
invention without authorization infringes the patent. 35
U.S.C. § 271(a) (2006). Yet, after eliminating the jury’s
lost profits award on these 18 scanners, the district court
failed to consider whether any “damages adequate to
compensate for the infringement” were owed to Siemens.
35 U.S.C. § 284 (2006); see also JMOL Opinion at 319. If
district court eliminates a lost profits award with regard
to a portion of infringing devices, the court must then
determine an appropriate measure of damages for that
portion. Crystal Semiconductor, 246 F.3d at 1355. Over-
looking the requirement of 35 U.S.C. § 284 is legal error.
Id.
    Here, the court abused its discretion by failing to de-
termine a reasonable royalty for the remaining 18 scan-
ners. We are not persuaded by Saint-Gobain’s arguments
that Siemens waived its right to reasonable royalty dam-
ages. The jury heard testimony from both parties’ dam-
ages experts on the subject of reasonable royalties. The
jury was also instructed on reasonable royalty damages.
In opposition to Saint-Gobain’s motion for JMOL, Sie-
33                           SIEMENS MEDICAL   v. SAINT GOBAIN


mens pointed out that a patent is infringed by making,
using, and selling the invention, and that the jury’s award
was likely a combination of both lost profits and a reason-
able royalty. Moreover, Saint-Gobain’s own damages
expert testified that, in the event that Saint-Gobain is
found to infringe, at least a reasonable royalty would be
owed on all 79 scanners, including the additional 18
scanners that were not sold by Philips. J.A. 1632-33.
    We therefore remand to the district court so that the
court may determine a reasonable royalty for the addi-
tional 18 scanners, to be added to the award of
$44,937,545 in lost profits for 61 scanners that were sold.
In determining the reasonable royalty, the district court
may rely on the present record if deemed adequate, or
may elect to receive additional expert testimony on what
royalty would be reasonable under the circumstances. 35
U.S.C. § 284 (2006).
                          CONCLUSION
    The district court’s judgment entered in favor of Sie-
mens is affirmed. However, we vacate the damages
award and remand for the court to consider a reasonable
royalty for the additional 18 infringing scanners, to be
added to the $44,937,545 in lost profits for the 61 scan-
ners that were sold.
     AFFIRMED IN PART, VACATED IN PART, AND
                   REMANDED
                            COSTS
      Costs to Siemens.
  United States Court of Appeals
      for the Federal Circuit
               __________________________

    SIEMENS MEDICAL SOLUTIONS USA, INC.,
            Plaintiff-Cross Appellant,

                            v.
  SAINT-GOBAIN CERAMICS & PLASTICS, INC.,
             Defendant-Appellant.
               __________________________

                    2010-1145, -1177
               __________________________

    Appeals from the United States District Court for the
District of Delaware in case no. 07-CV-0190, Judge Sue L.
Robinson.
                __________________________

PROST, Circuit Judge, dissenting.

    I respectfully dissent because, in my view, there is a
tension between the equivalence and nonobviousness
inquiries presented in this appeal.

    As it has here, this tension comes to light when a pat-
entee asserts the doctrine of equivalents against some
feature that the U.S. Patent and Trademark Office
(“PTO”) declared to be a point of novelty in a separate
patent. To prevail on equivalence, the patentee must
prove that the feature is insubstantially different from
the limitations of his asserted claim. Graver Tank & Mfg.
Co. v. Linde Air Prods. Co., 339 U.S. 605, 610 (1950);
SIEMENS MEDICAL   v. SAINT GOBAIN                        2


accord Chiuminatta Concrete Concepts, Inc. v. Cardinal
Indus., Inc., 145 F.3d 1303, 1310 (Fed. Cir. 1998). In an
ordinary case, we would say that the patentee’s efforts to
prove infringement will not be hindered by an accused
infringer’s proof of separate patenting. Atlas Powder Co.
v. E.I. du Pont De Nemours & Co., 750 F.2d 1569, 1581
(Fed. Cir. 1984). The equivalence analysis, however,
presents a unique circumstance.

    Where equivalence is used against a feature that is
separately patented, the patentee’s attempts to show that
the feature is insubstantially different from the asserted
patent crash into the well-settled presumption that
patents are nonobvious over the prior art. See Zygo Corp.
v. Wyko Corp., 79 F.3d 1563, 1570 (Fed. Cir. 1996). When
a fact finder concludes that a feature is “insubstantially
different” from a patent, how are we then to view the
validity of the patent putatively establishing that same
feature as “nonobvious”?

     Examining the legal standards underlying obvious-
ness and equivalence only emphasizes the potential for
overlap. The Supreme Court has instructed us that four
factors should be used in assessing obviousness: (1) the
scope and content of the prior art, (2) the differences
between the prior art and the claims, (3) the level of
ordinary skill in the art, and (4) secondary considerations
such as commercial success, unexpected results, and long-
felt need. Graham v. John Deere Co. of Kan. City, 383
U.S. 1, 17–18 (1966); accord Syntex (U.S.A.) LLC v. Apo-
tex, Inc., 407 F.3d 1371, 1378 (Fed. Cir. 2005). These
factors are intended to guide judges and juries in deter-
mining whether a hypothetical “skilled artisan” would
have viewed the claimed invention as one that was sug-
gested in a straightforward way by the prior art and the
general level of knowledge in the field. KSR Int’l Co. v.
3                           SIEMENS MEDICAL   v. SAINT GOBAIN


Teleflex Inc., 550 U.S. 398, 417–18 (2007). The caselaw is
replete with courts elaborating these standards in the
hope of making this difficult analysis as fair and straight-
forward as possible.

    The standard for equivalence—“insubstantial differ-
ence”—has also been well-explored. If a skilled artisan, at
the time of the accused infringement, viewed a substitu-
tion to a patented invention as insubstantially different
from the claim, the substitution is equivalent and in-
fringement may arise. Graver Tank, 339 U.S. at 610;
Warner-Jenkinson Co. v. Hilton Davis Chem. Co., 520
U.S. 17, 40 (1997). The doctrine prevents accused in-
fringers from evading liability by making “trivial” changes
to the patented invention. Festo Corp. v. Shoketsu Kin-
zoku Kogyo Kabushiki Co., 535 U.S. 722, 733 (2002). That
a substitution is “trivial” or “insubstantially different”
may be proved by showing that the substitution performs
the same function, in the same way, with the same result
as the technology of the claim limitation. Catalina Mar-
keting Int’l, Inc. v. Coolsavings.com, Inc., 289 F.3d 801,
813 (Fed. Cir. 2002).

    The majority reasons that these two tests, obvious-
ness and equivalence, are separate from one another. But
while the details differ, I find there is an inevitable area
of overlap. Assume a court, applying Graver Tank and its
progeny, found that to a person of skill in the art a substi-
tution was insubstantially different from a claim limita-
tion. Having so found, and setting aside (for the moment)
consideration of the time frames at which obviousness
and equivalence are assessed, the court would need only a
further finding that the skilled artisan had some reason
to make the substitution to find the limitation obvious
SIEMENS MEDICAL   v. SAINT GOBAIN                           4


under Graham and KSR. 1 This is not a high bar. For a
truly insubstantial change, the predictability of outcome
when substituting the one for the other suggests that a
reason to combine will be easy to prove. Rothman v.
Target Corp., 556 F.3d 1310, 1319 (Fed. Cir. 2009).

    I therefore disagree with the majority in that I do not
view it acceptable to leave the fact-finder—here, a jury—
without instruction on how it might navigate the equiva-
lence inquiry without undermining the presumption of
nonobviousness that we must accord issued patents.

    Judge Nies anticipated the challenge we face when
she wrote, “a substitution in a patented invention cannot
be both nonobvious and insubstantial.” Roton Barrier,
Inc. v. Stanley Works, 79 F.3d 1112, 1128 (Fed. Cir. 1996)
(Nies, J., Additional Views). A similar concern was voiced


        1    In the past, some litigants attempted to avoid
this second step entirely, and prove “anticipation by
equivalents.” This court rejected that theory, holding that
a claim is invalid under § 102 if and only if each and every
limitation of the claim is literally present in a single prior
art reference. But while it rejected “anticipation by
equivalents,” this court noted that the doctrine of equiva-
lents was, “if one wished to draw a parallel, . . . somewhat
akin to obviousness.” Lewmar Marine, Inc. v. Barient,
Inc., 827 F.2d 744, 748 (Fed. Cir. 1987); see also Richard-
son v. Suzuki Motor Co., 868 F.2d 1226, 1236 (Fed. Cir.
1989) (discussing the doctrine of equivalents as “a legal
theory that is pertinent to obviousness under Section 103,
not to anticipation under Section 102”). Notably, how-
ever, this court has declined to extend the doctrine of
equivalents to cover on infringement embodiments that
were present in the prior art. Wilson Sporting Goods Co.
v. David Geoffrey & Assocs., 904 F.2d 677, 685 (Fed. Cir.
1990), overruled in part on other grounds by Cardinal
Chem. Co. v. Morton Int’l, Inc., 508 U.S. 83, 99–103
(1993).
5                           SIEMENS MEDICAL   v. SAINT GOBAIN


by Judge Dyk (and quoted by the majority) in this court’s
Festo decision. Maj. Op. at 10 (quoting Festo Corp. v.
Shoketsu Kinzoku Kogyo Kabushiki Co., 493 F.3d 1368,
1379–80 (Fed. Cir. 2007)). This appeal now squarely
presents the issue.

    I believe that the best path through this difficult
situation is to expressly incorporate Judges Nies and
Dyk’s reasoning into the equivalence analysis: a sepa-
rately-patented (and presumptively nonobvious) substitu-
tion cannot be “insubstantial” unless some fact
distinguishes the equivalence finding from the PTO’s
earlier nonobviousness determination.

    The passage of time presents one opportunity for such
a distinction. The majority is correct that an equivalence
analysis differs from an obviousness analysis in the time
frame concerned. Obviousness is assessed at the time of
the invention, and equivalence at the time of the alleged
infringement. A finding that, from the time of the sepa-
rately-patented invention to the time of accused infringe-
ment, the state of the art advanced in such a way as to
make a previously non-obvious substitution obvious,
would allow an equivalence finding without upsetting the
presumption of nonobviousness. In the absence of such a
finding, in my view a separately-patented substitution
may not be held equivalent.

    Returning to the present appeal, I believe the trial
court should have given more direct guidance to the jury
concerning the effect the ’420 patent’s separate coverage
of Saint-Gobain’s 10% Y LYSO crystals had on the jury’s
duty as fact finder. Specifically, I believe it was reversible
error for the court not to inform the jury that it could find
equivalence if and only if it also found that the state of
the art concerning LSO crystals had advanced from the
SIEMENS MEDICAL   v. SAINT GOBAIN                        6


time of the ’420 patent’s invention to the alleged in-
fringement so that the 10% Y composition, though previ-
ously nonobvious, had become obvious and insubstantial. 2
I therefore respectfully dissent.




       2    In this appeal, Saint-Gobain sought a jury in-
struction emphasizing the ’420 patent’s presumption of
validity. I believe such an instruction should have been
given, and so dissent. But in my view the additional
guidance set forth herein was necessary to give full effect
to the instruction Saint-Gobain articulated.
