LOURIE, Circuit Judge.
This is a patent infringement suit involving two patents on memory modules. It comes to us as a consolidated appeal and cross-appeal from the August 23, 1991 judgment, 1991 WL 333696, and the October 23, 1991 amended judgment of the United States District Court for the Eastern District of Virginia, Civil No. 90-1477-A. On August 8,1991, a jury returned a verdict in favor of Wang Laboratories, Inc., finding U.S. Patents 4,656,605 and 4,727,513 infringed by Toshiba Corporation, Toshiba America Electronic Components, Inc., and Toshiba America Information Systems, Inc. (collectively Toshiba) and NEC Corporation, NEC Electronics Inc., and NEC Technologies, Inc. (collectively NEC), and not invalid. The jury found that NEC had willfully infringed the patents in suit, while Toshiba’s infringement was not willful. On August 14, 1991, the jury determined that Wang was entitled to a reasonable royalty as compensation for the infringement and determined alternative reasonable royalty rates. On August 23, 1991, the district court entered judgment in accordance with the jury verdicts, ruled that the patents were not invalid and were infringed, and awarded a reasonable royalty as damages.
Toshiba and NEC filed post-trial motions for judgment notwithstanding the verdict (JNOV),1 which the district court subsequently denied.2 Wang moved to amend the judgment as to damages; this motion was also denied. Toshiba and NEC now appeal from the district court’s judgment denying JNOV on the issues of validity and infringement. Wang cross-appeals the district [861]*861court’s failure to amend the judgment as to the royalty rate. We affirm-in-part, reverse-in-part, and remand-in-part.
BACKGROUND
The ’605 and ’513 patents, both entitled “Single In-Line Memory Module,” were issued in the name of James E. Clayton and assigned to Wang. The ’605 patent issued on April 7, 1987, and the ’513 patent on February 23,1988 from a continuation of the application that led to the ’605 patent.3
The patents relate to single in-line memory modules (SIMMs) having eight data memory chips capable of storing 8-bit binary words or bytes.4 Additionally, the memory modules include a ninth chip, which functions as a check or parity bit for error detection.5 The nine memory chips, which are packaged in plastic leaded chip carriers (PLCCs), are mounted on a single epoxy-glass printed circuit board substrate. Decoupling capacitors for suppressing voltage spikes are also mounted on the memory module substrate. Preferably, access terminals are arrayed across the bottom of the device for data input and output, data address and memory control, and device power. The ’605 patent claims require that the ninth chip be interconnected with the other eight, while the ’513 patent claims do not require this interconnection so that the parity chip can be written to or read from independently of the eight data chips.
The ’605 patent contains one claim, which reads as follows:
A memory module for installation on a printed circuit motherboard comprising:
eight data memory chips for storing digital data, each having a data input and output, a control input, and an address input, and each being packaged in a plastic leaded chip carrier;
a ninth memory chip for storing error detection and correction information associated with the eight data memory chips, said ninth memory chip having a data input and output, a control input and an address input interconnected with those of the eight memory chips, and a control input to provide writing in or reading out of the ninth memory chip at times other than when said bytes of digital information are written into or read out of the eight data memory chips to thereby facilitate said error detection and correction operation;
an epoxy-glass printed circuit hoard substrate having a length and width adequate for mounting thereon only in a single row said nine memory chips and for interconnecting the control inputs and the address inputs of the memory chips so that bytes of digital information may be input to or output from the memory chips one at a time;
the substrate including thirty terminals for providing access to the data inputs and outputs, control inputs, and address inputs of the nine memory chips to enable reading and writing of bytes of digital information into and out of the eight memory chips and to enable reading and writing of error detection and correction information into and out of the eight memory chips;
support means for supporting the memory module at an angle with respect to the printed circuit motherboard when the memory module is installed thereon; and eight decoupling capacitors, mounted on said substrate and connected between the nine memory chips, for suppressing transient voltage spikes between said memory chips.
(Emphasis added).
The ’513 patent contains Claims 1 and 2, which read as follows:
[862]*8621. A memory module for installation on a printed circuit motherboard comprising
nine data memory chips for storing digital data, each having a data input and output, control input, and an address input, and each being packaged in a plastic leaded chip carrier, wherein said ninth memory chip is for storing detection and correction information associated with the eight data memory chips,
an epoxy-glass printed circuit board substrate having a length and width adequate for mounting thereon only in a single row said nine memory chips and for interconnecting the control inputs and the address inputs of the memory chips so that bytes of digital information may be input to or output from the memory chips,
the substrate including thirty terminals for providing access to the data inputs and outputs, control inputs, and address inputs of the nine memory chips and to enable reading and writing of information into and out of the nine chips,
support means for supporting the memory module at an angle with respect to a motherboard and
decoupling capacitors mounted on said substrate and coupled to the memory chips for suppressing transient voltages.
2. The module of claim 1 wherein all nine memory chips are interconnected such that data is input to or output from the ninth memory chips when data is input to or output from the other eight memory chips.
Toshiba and NEC manufacture several different types of memory modules, including (1) classic nine-chip modules, which have eight data chips and one error detection chip arranged in a single row on a printed circuit board substrate; (2) 3-pack modules, which have three chips arranged in a single row, two half-bytes, which read and store four bits of information each, and a parity chip; and (3) lateral modules, which have nine chips arranged in more than a single row. Of these modules, some are leaded, i.e., electrical leads extend from the module substrate and electrically connect the module to a printed circuit motherboard, and some are leadless, i.e., an edge of the module is designed to mate with a socket attached to a printed circuit motherboard. The jury found, inter alia, that the classic modules literally infringed claim 1 of the ’513 patent and infringed claim 1 of the ’605 patent and claim 2 of the ’513 patent under the doctrine of equivalents; the 3-pack modules (all lead-less) infringed claim 1 of both patents under the doctrine of equivalents; and the lateral modules infringed claim 1 of both patents under the doctrine of equivalents.6
The jury determined two reasonable royalty rates, one 4.0%, assuming hypothetical royalty negotiations to have occurred in April 1987, the date when the ’605 patent issued, and the other 2.75%, assuming hypothetical royalty negotiations to have occurred in January 1990, the date Wang gave notice of infringement. The court adopted January 1990 as the date when hypothetical royalty negotiations occurred, making the royalty rate 2.75%. The parties stipulated to the amount of infringing sales, excluding sales covered by a licensing agreement between Wang and IBM, as being $31,106,509 by NEC and $88,121,819 by Toshiba. The court awarded double damages to Wang for NEC’s willful infringement. Based on a 2.75% royalty rate, Wang was awarded $855,429 to be paid by NEC and $2,423,350 to be paid by Toshiba, plus prejudgment interest of 8.0% per annum from October 1, 1990 to August 23, 1991, the date of entry of judgment, plus post-judgment interest at the statutory rate. The court also issued a permanent injunction against both Toshiba and NEC.7
[863]*863Toshiba and NEC jointly moved for JNOV on the issues of best mode, written description, obviousness, and infringement, all of which were denied. NEC moved for JNOV on the issues of willfulness, which was denied, and enhanced damages, which was granted. Wang moved to amend the judgment as to the date of the hypothetical negotiations, to increase the enhanced damages, and for attorney fees, all of which were denied. Toshiba and NEC now appeal from the district court’s judgment denying JNOV on the issues of obviousness, failure to meet the written description requirement, and infringement. Wang cross-appeals the court’s failure to amend the judgment as to the reasonable royalty rate.
DISCUSSION
I. Standard of Review from Denial of JNOV on Infringement and Validity
On appeal of a judgment entered on a verdict after denial of a motion for JNOV, Toshiba and NEC must show
(1) that reasonable persons could not in light of [] evidence [before them] have found the facts necessary to support the jury’s verdict; or (2) that the facts properly found cannot in law support that verdict.
Railroad Dynamics, Inc. v. A. Stucki Co., 727 F.2d 1506, 1513, 220 USPQ 929, 936 (Fed.Cir.), cert. denied, 469 U.S. 871, 105 S.Ct. 220, 83 L.Ed.2d 150 (1984). Fact findings reviewed under the substantial evidence standard require affirmance unless appellants show that no reasonable juror could have reached such a result. Id. In reviewing the evidence from a denial of JNOV, we must
(1) consider all the evidence, (2) in a light most favorable to the non-mover[,] (3) drawing reasonable inferences favorable to the non-mover[,] (4) without determining credibility of witnesses, and (5) without substituting [our] choice for that of the jury between conflicting elements in the evidence.
Dana Corp. v. IPC Ltd. Partnership, 860 F.2d 415, 417, 8 USPQ2d 1692, 1694-95 (Fed.Cir.1988) (citations omitted), cert. denied, 490 U.S. 1067, 109 S.Ct. 2068, 104 L.Ed.2d 633 (1989).
II. Obviousness
Toshiba and NEC argue that the claims in suit are invalid for obviousness under 35 U.S.C. § 103 (1988).8 Specifically, they state that the claimed subject matter would have been obvious in view of U.S. Patent 4,281,392 assigned to Allen-Bradley Company, sales of Allen-Bradley’s X9 SIMMs, and Texas Instruments 1982 MOS Memory Data Book. The appellants further assert that the district court erroneously read limitations into the claims, viz., use of the memory modules in a “personal computer”; attributes of greater storage capacity and less cost for the memory modules compared with those in the prior art; and possession of exactly 30 terminals. While we agree that the claims do not contain these limitations, we do not find this to be determinative on the question of nonob-viousness. There is substantial evidence that certain of the prior art was not analogous, and hence the validity of the claims can be upheld for that reason.
Nonobviousness is a legal conclusion, having factual underpinnings, including the scope and content of the prior art. Graham v. John Deere Co., 383 U.S. 1, 17, 86 S.Ct. 684, 693, 15 L.Ed.2d 545, 148 USPQ 459, 467 (1966). “When the Graham factual underpinnings have been genuinely disputed, as in this case, we presume that the jury resolved them in favor of the verdict winner.” Jurgens v. McKasy, 927 F.2d 1552, 1558, 18 USPQ2d 1031, 1036 (Fed.Cir.) (citing Perkin-Elmer Corp. v. Computervision Corp., 732 F.2d 888, 893, 221 USPQ 669, 674 (Fed.Cir.), cert. denied, 469 U.S. 857, 105 S.Ct. 187, 83 L.Ed.2d 120 (1984)), cert. denied, - U.S. -, 112 S.Ct. 281, 116 L.Ed.2d 232 (1991); see also Newell Cos. v. Kenney Mfg. Co., 864 F.2d 757, 765, 9 USPQ2d 1417, 1423 (Fed.Cir.1988) (“Judges must accept the factual findings, presumed from a favorable jury [864]*864verdict, which are supported under the substantial evidence/reasonable juror standard.”), cert. denied, 493 U.S. 814, 110 S.Ct. 62, 107 L.Ed.2d 30 (1989).
Appellants assert that Allen-Bradley’s ’392 patent and its commercial counterpart, the X9 SIMM, are analogous to the claimed subject matter, and accordingly that they are effective to render the claims in suit invalid. However, because of the adequate jury instruction concerning analogous art, we will presume that the Allen-Bradley art was found to be non-analogous to the claimed subject matter. The question then is whether that finding is supported by substantial evidence. We conclude that there was substantial evidence.
Analogous art is that which is relevant to a consideration of obviousness under section 103. See In re Sovish, 769 F.2d 738, 741, 226 USPQ 771, 773 (Fed.Cir.1985). “Whether something legally within the prior art is ‘analogous’ is a fact question.... ” Panduit Corp. v. Dennison Mfg. Co., 810 F.2d 1561, 1568 n. 9, 1 USPQ2d 1593, 1597 n. 9 (Fed.Cir.), cert. denied, 481 U.S. 1052, 107 S.Ct. 2187, 95 L.Ed.2d 843 (1987). Two criteria are relevant in determining whether prior art is analogous: (1) whether the art is from the same field of endeavor, regardless of the problem addressed, and (2) if the art is not within the same field of endeavor, whether it is still reasonably pertinent to the particular problem to be solved. In re Clay, 966 F.2d 656, 658-59, 23 USPQ2d 1058, 1060 (Fed.Cir.1992) (citations omitted).
The ’392 patent is entitled “Memory Circuit for Programmable Machines”; it discloses a SIMM containing nine memory chips, eight for storing data and one for error detection, mounted in a single row. In the late 1970’s, Allen-Bradley manufactured and sold the X9 SIMM for use in its 9-bit programmable controller. This product consisted of nine memory chips encapsulated in ceramic dual in-line packages (ceramic DIPs) mounted on an epoxy-glass printed circuit board substrate.
The Allen-Bradley art is not in the same field of endeavor as the claimed subject matter merely because it relates to memories. It involves memory circuits in which modules of varying sizes may be added or replaced; in contrast, the subject patents teach compact modular memories. Thus, based on the evidence of record, the jury could reasonably have found that the first criterion of the analogous art test has not been met and that the prior art and the claimed subject matter are not in the same field of endeavor.
Even though the Allen-Bradley art is not within the relevant field of endeavor, it may still be analogous if it is reasonably pertinent to the problem the inventor attempted to solve. Id. at 659, 23 USPQ2d at 1060-61 (citation omitted). “A reference is reasonably pertinent if, even though it may be in a different field from that of the inventor’s endeavor, it is one which, because of the matter with which it deals, logically would have commended itself to an inventor’s attention in considering his problem.” Id. at 659, 23 USPQ2d at 1061. However, given the jury’s ultimate conclusion, we presume that the jury decided that the Allen-Bradley art was not reasonably pertinent. The question then is whether that conclusion is supported by substantial evidence.
Dr. Jeffrey Frey, Wang’s technical expert, testified that the Allen-Bradley technology, including the SIMM described in the patent and the X9, was not pertinent to the field of personal computers for which Wang’s SIMMs were designed. Although Wang’s patents do not mention the term “personal computer,” Dr. Frey stated that “[t]he entire context of the patent[s] — in the application of the memories, units of nine, dynamic memories — indicates they’re meant for use in personal computers.” Dr. Frey further testified that the Allen-Bradley module was developed for use in a controller of large industrial machinery and could not be used in a personal computer. He also stated that the Allen-Bradley patent teaches the use of Static Random-Access-Memories (SRAMs) or Read-Only-Memories (ROMs) and does not suggest the use of Dynamic Random-Aceess-Memories (DRAMs) as taught by Wang. As Dr. Frey stated, DRAMs are primarily used in personal computers (PCs), while SRAMs, which are larger and more expensive, are not used in PCs.
[865]*865Wang’s SIMMs were designed to provide compact computer memory with minimum size, low cost, easy repairability, and easy expandability. See ’605 patent, col. 2, lines 61-64 (“By using the small D-RAMs and small capacitors, module 30 may have physical dimensions [on] the order of three-quarter inch by three inches while providing large memory capacity.”). In contrast, the Allen-Bradley patent relates to a memory circuit for a larger, more costly industrial controller. SRAMs were used by Allen-Bradley because of their intended industrial environment. According to Dr. Frey, size was not a consideration in the Allen-Bradley work. Thus, there is substantial evidence in the record to support a finding that the Allen-Bradley pri- or art is not reasonably pertinent and is not analogous.
Toshiba and NEC also argue that Wang stipulated that the Allen-Bradley art is analogous. We disagree. At trial, the parties stipulated that the Texas Instruments 1982 MOS Memory Data Book, the ’392 patent, and Allen-Bradley’s X9 SIMM were prior art to the ’605 and ’513 patents, but were not before the examiner during prosecution. When the court asked if “[t]hese are the stipulated prior art references[?],” and referred to the Allen-Bradley patent and X9 SIMM among other art, Wang did not stipulate that this art was analogous. Wang acknowledged only that it was prior art, not analogous prior art.
The Allen-Bradley patent and X9 SIMM, not being analogous prior art, thus could not have rendered the claimed subject matter obvious. See Jurgens, 927 F.2d at 1559, 18 USPQ2d at 1036. Since Toshiba and NEC rely principally on the Allen-Bradley art and only point to the Texas Instruments reference in combination with Allen-Bradley, appellants have failed to show that the claimed subject matter would have been obvious to a person having ordinary skill in the art at the time Wang’s inventions were made.9
III. Written Description
NEC10 argues that the ’605 and ’513 patents are invalid for failure to comply with the written description requirement of 35 U.S.C. § 112, first paragraph, which states that
[t]he specification shall contain a written description of the invention, and of the manner and process of making and using it, in such full, clear, concise, and exact terms as to enable any person skilled in the art ...
35 U.S.C. § 112 (1988) (emphasis added). The standard for determining whether the written description requirement has been met has been stated as follows:
Although [the applicant] does not have to describe exactly the subject matter claimed, ... the description must clearly allow persons of ordinary skill in the art to recognize that [he or she] invented what is claimed.... The test for sufficiency of support in a parent application is whether the disclosure of the application relied upon reasonably conveys to the artisan that the inventor had possession at that time of the later claimed subject matter.
Vas-Cath Inc. v. Mahurkar, 935 F.2d 1555, 1563, 19 USPQ2d 1111, 1116 (Fed.Cir.1991) (citations and quotations omitted). Whether the written description requirement has been met is a question of fact. Ralston Purina Co. v. Far-Mar-Co. Inc., 772 F.2d 1570, 1575, 227 USPQ 177, 179 (Fed.Cir.1985). Thus, we must determine whether substantial evidence supports the verdict that the written description requirement was met.
According to NEC, the recitation of “support means for supporting the memory mo[866]*866dule at an angle with respect to the printed circuit motherboard” was not supported by the original Clayton application, and therefore the claims in both patents are invalid. NEC notes that “support means” was not recited in the original claims, but was added by amendment. It alleges that the specification describes the leads of the memory modules as having only an electrical function, not a mechanical, support function.
A patent specification is directed to one of ordinary skill in the art. In re Hayes Microcomputer Prods., Inc. Patent Litigation, 982 F.2d 1527, 1533, 25 USPQ2d 1241, 1245 (Fed.Cir.1992). It is also clear that “drawings alone may provide a ‘written description’ of an invention as required by § 112.” Vas-Cath, 935 F.2d at 1565, 19 USPQ2d at 1118. Dr. Frey testified that Figure 2 “show[s] terminals as leads, which are means of supporting the module.” Additionally, Dr. Frey stated, when discussing whether leadless SIMMs are disclosed in the patents, that a person of ordinary skill in the art would know that a leadless SIMM includes a row of terminals “to mount and support that module.” He went on to state that “it’s the edge of the card and the terminals that support the module.” The inventor, Mr. Clayton, also testified that on leaded SIMMs, the leads themselves are the support means, and that on leadless SIMMs, the bottom row of terminals constitutes the support means. Thus, there is substantial evidence in the record to support the conclusion that the support means element is adequately described in the specification by the disclosure of both leads and the terminal edge of the modules. NEC has not shown that the district court’s denial of JNOV on the issue of failure to meet the written description requirement was incorrect.
IV. Infringement
A. Lateral and 3-Pack Memory Modules
Toshiba and NEC both appeal that part of the judgment finding infringement under the doctrine of equivalents by the sale of their lateral and 3-pack memory modules. Toshiba argues that prosecution history estoppel bars application of the doctrine. Specifically, Toshiba argues that the 3-pack modules have three memory chips rather than nine, and the lateral modules do not have memory chips mounted only in a single row. They assert that both of these claim limitations were added to overcome rejections based on prior art. Accordingly, Toshiba argues, Wang is now estopped from claim scope given up during prosecution.
Although Toshiba asserts that prosecution history estoppel applies to both its lateral and its 3-pack memory modules, NEC only raises estoppel with respect to its 3-pack modules and argues that its lateral modules do not infringe Wang’s claims because they are leadless. Normally an issue not raised by an appellant in its initial brief is waived. However, “[t]his practice is ... not governed by a rigid rule but may as a matter of discretion not be adhered to where circumstances indicate that it would result in basically unfair procedure.” Becton Dickinson & Co. v. C.R. Bard, Inc., 922 F.2d 792, 800, 17 USPQ2d 1097, 1103 (Fed.Cir.1990). In this case, we can hardly give one appellant the benefit of an estoppel without giving it to the other, so we consider the defense of prosecution history estoppel as applicable to both Toshiba and NEC for both the 3-pack and lateral modules. Wang is not prejudiced by this because it was afforded the opportunity at trial and on appeal to address the issue with respect to the lateral modules. See Singleton v. Wulff, 428 U.S. 106, 120, 96 S.Ct. 2868, 2877, 49 L.Ed.2d 826 (1976). Moreover, in this case both parties relied on an estoppel defense at trial.
Prosecution history estoppel bars “a patentee from enforcing its claims against otherwise legally equivalent structures if those structures were excluded by claim limitations added in order to avoid prior art.” Mannesmann Demag Corp. v. Engineered Metal Prods. Co., 793 F.2d 1279, 1284, 230 USPQ 45, 48 (Fed.Cir.1986) (citations omitted). “In determining whether prosecution history estoppel applies because of a change in claim language during prosecution, the court must consider not only what was changed, but the reason for such change.” Vaupel Textilmaschinen KG v. Meccanica Euro Italia S.P.A., 944 F.2d 870, 882, 20 [867]*867USPQ2d 1045, 1054 (Fed.Cir.1991) (citing Sun Studs, Inc. v. ATA Equip. Leasing, Inc., 872 F.2d 978, 987, 10 USPQ2d 1338, 1345 (Fed.Cir.1989)). Whether estoppel applies is a question of law. LaBounty Mfg., Inc. v. United States Int’l Trade Comm’n, 867 F.2d 1572, 1576, 9 USPQ2d 1995, 1998 (Fed.Cir.1989).
Toshiba asserts that Wang specifically limited the scope of its claims to nine memory chips contained in a single row in order to overcome prior art and that Wang is precluded from recapturing what it gave up during prosecution. “Unmistakable assertions made by the applicant to the Patent and Trademark Office (PTO) in support of patentability, whether or not required to secure allowance of the claim, also may operate to preclude the patentee from asserting equivalency_” Texas Instruments Inc. v. United States Int’l Trade Comm’n, 988 F.2d 1165, 1174, 26 USPQ2d 1018, 1025 (Fed.Cir.1993) (citations omitted). “Depending on the nature and purpose of an amendment, it may have a limiting effect within a spectrum ranging from great to small to zero.” Hughes Aircraft Co. v. United States, 717 F.2d 1351, 1363, 219 USPQ 473, 481 (Fed.Cir.1983). The prosecution history must be examined as a whole in determining whether estoppel applies. Texas Instruments, 988 F.2d at 1174, 26 USPQ2d at 1025.
In the first Office Action in Wang’s parent application,11 the claims were rejected under 35 U.S.C. § 103 as unpatentable over U.S. Patent 3,972,033 to Cislaghi, who teaches a memory module with multiple rows of nine chips. In Cislaghi the check bit, the ninth chip, is located on a separate module from the other eight bits. In response to the rejection, Wang replaced the filed claims so that they recited a printed circuit board substrate for mounting a plurality of memory chips in a single row. Wang stated:
The concept of applicant’s invention lies in an improved memory module for installation on a printed circuit memory board. A single row of RAM memory chips, packaged in the plastic leaded chip carriers, is mounted ón a printed circuit board (glass-epoxy) substrate....
Cislaghi et al. discloses a memory consisting of two “submodule” printed circuit cards wherein check bits for the first card are contained on the second card and vice versa. Applicant’s invention as presently claimed has little in common with the disclosure of Cislaghi.... Cislaghi [does not] have memory chips mounted in a single row....
Thus, Wang added claims which recited “mounting said plurality of memory chips thereon in a single row.” (Emphasis added). The limitation of mounting chips “in a single row” was a basis for overcoming the Examiner’s rejection based on Cislaghi. Unlike the teaching of Cislaghi, Wang asserted that its invention accommodated the size constraints of its memory module, noting “[t]he purpose of applicant’s invention is to conserve space on a memory board.” The “single row” limitation was clearly made for the purpose of overcoming the prior art.
After a second Office Action, Wang filed a preliminary amendment in a file wrapper continuation application in which it cancelled its previous claims and added a new claim, reciting “an epoxy-glass printed circuit board substrate having a length and width adequate for mounting thereon only in a single row said nine memory chips.” (Emphasis added). In the “Remarks” section accompanying the preliminary amendment, Wang stated that “Cislaghi et al. teaches the advantage of locating a parity chip on a different printed circuit board.... ” Once again, these limitations were added to overcome the Examiner’s rejection based on Cislaghi.
Wang argues that the prosecution history does not exclude a substrate having memory chips mounted in two rows, but that the prior art only restricts the size of the substrate to one no larger than that on which nine DRAMs can be mounted. While a two-row construction may not read on the prior art, Wang limited the scope of its claims to memory chips mounted “only in a single row,” and [868]*868twice argued before the PTO that Cislaghi did not have memory chips mounted in a single row. Although Cislaghi taught the advantage of locating a parity chip on a separate module, Wang did not limit its claims to a single module. Instead, it chose a single row, which excludes the accused modules of Toshiba and NEC. Wang chose to emphasize the specific dimensions of the printed circuit board taught by the invention and limited itself to exclude any embodiment in which the memory chips are not physically located in a single row. See Litton Sys., Inc. v. Whirlpool Corp., 728 F.2d 1423, 1439, 221 USPQ 97, 107 (Fed.Cir.1984) (“A patent attorney is often faced with choices during a patent prosecution.... A patent attorney should not be able, however, to choose one course of action within the PTO with the anticipation that, if later checked, he or she can always choose an alternate course of prosecution in a trial before a federal judge.”). Thus, the court erred in denying JNOV; prosecution history estoppel should have been applied to reach the conclusion that the accused lateral memory modules were surrendered during prosecution and therefore do not infringe under the doctrine of equivalents.
Likewise, Wang is also estopped from now arguing that the 3-pack memory modules having less than nine chips can infringe its claims. In the second Office Action, independent claim 4 was rejected under 35 U.S.C. § 103 as unpatentable over a publication by Electronic Designs Inc. (EDI) in view of an article by Lowe. The Examiner stated that the EDI publication showed all the elements of the recited claims except the use of PLCCs and that this technique was shown in Lowe. The EDI publication disclosed a memory module having eight chips and decoupling capacitors in which the module was leaded.
In response, Wang replaced its claims which recited “a plurality of data memory chips” with a new claim reciting “eight data memory chips” and “a ninth memory chip for storing error detection and correction information.” In the “Remarks” section of its amendment, Wang argued that the EDI publication described “a single in-line memory module which consists of eight ceramic chips- Applicant claims a single in-line memory module comprising nine memory chips (eight for data, one for parity)_” (Emphasis added). Wang also argued that “Cislaghi et al. is opposite to the teaching of the present invention, which claims ... a ninth memory chip.... ” (Emphasis added).
Wang now argues that the prosecution history does not exclude use of three memory chips which function equivalently to nine memory chips. However, Wang limited the scope of its claims to nine chips. During prosecution, Wang distinguished the EDI publication, which taught the use of eight chips, by adding a claim reciting exactly nine chips. Wang is thus precluded from obtaining the benefit of the doctrine of equivalents for 3-pack memory modules. The court accordingly erred in denying JNOV as to that issue.
We therefore reverse that part of the court’s judgment finding infringement by equivalence as applied to the lateral and 3-pack memory modules. Since the same es-toppel exists with respect to both the ’605 and ’513 patents, our ruling applies to both patents.
B. Classic, Leadless Memory Modules
NEC also appeals the denial of JNOV respecting the jury’s verdict that NEC’s leadless memory modules infringe the patents in suit.12 NEC argues generally that none of its leadless modules infringes literally or under the doctrine of equivalents. NEC specifically states that the claims require a “support means” to hold the module at an angle to the motherboard and that the jury misconstrued the claims in finding that NEC’s leadless modules infringe.
In response, Wang states that the sufficiency of the evidence underlying the district court’s denial of JNOV on this issue is unre-viewable on appeal since NEC failed to move for directed verdict on this ground.13 NEC [869]*869responds that it did move for summary judgment. We agree with Wang. Although it is true that the standard for summary judgment is virtually the same as that for a directed verdict, viz., that “there can be but one reasonable conclusion as to the verdict,” Anderson v. Liberty Lobby, Inc., 477 U.S. 242, 250, 106 S.Ct. 2505, 2511, 91 L.Ed.2d 202 (1986) (citation omitted), this does not mean that a motion for summary judgment is a substitute for a motion for directed verdict. Rule 50(b) of the Federal Rules of Civil Procedure requires a motion for a directed verdict as a prerequisite to a motion for JNOV. A motion for summary judgment before trial is not a substitute for a motion for directed verdict at the close of all the evidence.
NEC also asserts that on an issue of claim construction we need not review the sufficiency of the evidence, but can decide as a matter of law whether a leadless memory module can satisfy the support means element of the claims. We disagree. What is involved here are factual questions, not questions of law. As discussed earlier, the leads on the leaded SIMMs are in fact disclosed as support means in the specification. Whether leadless memory modules infringe the claims in issue is also a question of fact. See Palumbo v. Don-Joy Co., 762 F.2d 969, 975, 226 USPQ 5, 8 (Fed.Cir.1985) (citing D.M.I., Inc. v. Deere & Co., 755 F.2d 1570, 1575, 225 USPQ 236, 239 (Fed.Cir.1985)). Since NEC did not raise this factual issue in a timely manner, we are precluded from reviewing the sufficiency of the evidence supporting the jury’s conclusion. This result also applies to both patents asserted and to arguments of literal infringement and the doctrine of equivalents.
V. Damages
The district court awarded damages to Wang based on a stipulated total of infringing sales and a reasonable royalty rate of 2.75%, assuming hypothetical royalty negotiations to have occurred in January 1990, the date Wang gave notice to Toshiba and NEC that their products infringed Wang’s patents, rather than in April 1987, when the ’605 patent issued. The district court stated that January 1990
is chosen because plaintiff is not entitled to any royalty damages prior to that date and because it is the date on which the parties actually would have engaged in negotiations over a reasonable royalty, had both sides been willing to do so. Hence, selection of this date best approximates the royalty that would have resulted had the defendants chosen to seek a patent license rather than to continue unlicensed production and risk being found to have infringed the patents. Nor is there any solid policy reason supporting selection of the earlier date, the date on which the patents issued and infringement of the invention technically began, but for which plaintiffs are barred by 35 U.S.C. § 287 from seeking damages.
Slip op. at 3.
Wang cross-appeals the district court’s selection of the January 1990 date as the date when hypothetical negotiations were presumed to have occurred. Wang argues that negotiations should have been hypothesized at the start of infringement, i.e., when both a patent had issued and accused products were sold. We agree.
In reviewing the district court’s award, we must determine if the court abused its discretion in its methodology for determining a reasonable royalty rate. SmithKline Diagnostics, Inc. v. Helena Lab. Corp., 926 F.2d 1161, 1164, 17 USPQ2d 1922, 1925 (Fed.Cir.1991). An abuse of discretion occurs when a court bases its determination “on clearly erroneous factual findings, legal error, or a manifest error of judgment.” Datascope Corp. v. SMEC, Inc., 879 F.2d 820, 823-24, 11 USPQ2d 1321, 1323 (Fed.Cir.1989), cert. denied, 493 U.S. 1024, 110 S.Ct. 729, 107 L.Ed.2d 747 (1990) (citation omitted). We conclude that the district court committed legal error in choosing January 1990 as the date when hypothetical negotiations began and therefore that it abused its discretion by denying Wang’s motion to [870]*870amend the judgment as to the reasonable royalty rate.
Under 35 U.S.C. § 284 (1988), the court shall award the claimant damages adequate to compensate for the infringement but in no event less than a reasonable royalty for the use made of the invention by the infringer....
When actual damages cannot be adequately proved, a reasonable royalty may be employed. Fromson v. Western Litho Plate & Supply Co., 853 F.2d 1568, 1574, 7 USPQ2d 1606, 1612 (Fed.Cir.1988). “A reasonable royalty is the amount that ‘a person, desiring to manufacture [, use, or] sell a patented article, as a business proposition, would be willing to pay as a royalty and yet be able to make [, use, or] sell the patented article, in the market, at a reasonable profit.’ ” Trans-World Mfg. Corp. v. Al Nyman & Sons, Inc., 750 F.2d 1552, 1568, 224 USPQ 259, 269 (Fed.Cir.1984) (alterations in original; citations omitted). When an established royalty does not exist, a court may determine a reasonable royalty based on “hypothetical negotiations between willing licensor and willing licensee.” Fromson, 853 F.2d at 1574, 7 USPQ2d at 1612. “The key element in setting a reasonable royalty ... is the necessity for return to the date when the infringement began.” Hanson v. Alpine Valley Ski Area, Inc., 718 F.2d 1075, 1079, 219 USPQ 679, 682 (Fed.Cir.1983) (quoting Panduit Corp. v. Stahlin Bros. Fibre Works, Inc., 575 F.2d 1152, 1158, 197 USPQ 726, 731 (6th Cir.1978)); see also Fromson, 853 F.2d at 1575, 7 USPQ2d at 1613 (hypothetical royalty negotiations methodology “speaks of negotiations as of the time infringement began”).
In choosing the January 1990 date, the district court failed to follow our precedent. It is not illogical to hypothesize a negotiation at the time of notice. After all, an accused infringer may not know of the patents until notice is given. Nonetheless, this case is governed by the rule in Fromson, in which hypothetical negotiations were determined to have occurred when the infringement began, which was the date the patent issued, even though, under 35 U.S.C. § 286, the infringer was only liable for damages for the six years prior to the filing of the infringement action. In this case, infringing products were being sold on the date of issuance of the ’605 patent. Therefore, under Fromson, hypothetical royalty negotiations should have been considered to have occurred on the patent issuance date. It is true that limitations may apply to the period for which damages may be recovered. As in the present case, failure to mark patented goods is a limitation on recovery of damages, in the absence of notice. 35 U.S.C. § 287 (1988). However, the court confused limitation on damages due to lack of notice with determination of the time when damages first began to accrue, and it is the latter which is controlling in a hypothetical royalty determination. Thus, the district court abused its discretion by denying Wang’s motion to amend the judgment as to the reasonable royalty rate.
CONCLUSION
We reverse the district court’s judgment denying NEC’s and Toshiba’s joint motion for JNOV on the issue of infringement by NEC’s and Toshiba’s sale of lateral and 3-pack memory modules under the doctrine of equivalents. We reverse the court’s judgment awarding damages at a 2.75% royalty rate, and we remand for a determination of damages at a 4.0% royalty rate. We affirm the judgment in all other respects.
COSTS
Each party is to bear its own costs.
AFFIRMED-IN-PART, REVERSED-IN-PART, AND REMANDED-IN-PART.