*662Following a trial, a jury returned a verdict for Timothy Larkin, who brought this medical malpractice suit individually and on behalf of his wife, Andrea Larkin,3 and his daughter, Alexa Larkin (collectively, plaintiffs, or Larkin). The jury awarded damages in the sum of $35.4 million, which a judge of the Superior Court later reduced to approximately $32.5 million on the motion of the defendant, Dedham Medical Associates (DMA). On appeal, DMA claims the judge made erroneous rulings on its posttrial motions. We affirm.4
Background. The unfortunate facts of this case are not disputed in this appeal. Andrea, a former elementary school teacher, exercise class instructor, and marathon runner, complained of persistent dizziness beginning in 2004. She underwent a magnetic resonance imaging procedure at Massachusetts General Hospital (MGH), which revealed a venous varix5 in her brain that was determined not to be a cause of her dizziness symptoms. Her *663primary care doctor, Jehane Johnston, an employee of DMA, was copied on the MGH report, but failed to note the venous varix on Andrea's "problem list," which is designed to alert a patient's various treating physicians to her medical conditions. As a result, when Andrea became pregnant in 2007, her obstetricians were not informed about her venous abnormality. Andrea was not informed that this condition posed a particular risk of the venous varix rupturing during vaginal labor using the Valsalva maneuver or that an elective Caesarian section would avoid placing additional stress on the veins in her head and neck.6 Andrea delivered her daughter *1216vaginally using the Valsalva maneuver in June of 2008.
Twelve hours after Andrea delivered her baby, she experienced a sudden, very painful headache, which accompanied a rupture of the venous varix in her brain. After having emergency surgery to remove a part of her skull and falling into a month-long coma, Andrea awoke to find that her legs and left hand were paralyzed, her trunk muscles were impaired, and that she had difficulty chewing, swallowing, and speaking. Despite having physical therapy twice per week, the occasional occupational, water, and speech therapies, and a "battery" of daily medications, Andrea's injuries are permanent. She requires care around the clock and likely will for the duration of her lifetime.
Larkin filed this suit against DMA and Dr. Johnston. A two-week long trial commenced on April 27, 2015. The jury rendered a verdict in favor of Larkin and awarded $35.4 million.7 DMA filed a series of posttrial motions for judgment notwithstanding the verdict, for a new trial and/or remittitur, to amend the pleadings to conform to the evidence, and to alter or amend the judgment, challenging the verdict and the jury award, alleging, among other things, (1) that it was entitled to the benefit of the statutory limitation on tort liability afforded to certain charitable organizations pursuant to G. L. c. 231, § 85K, as then in effect, (2) that testimony provided by Larkin's expert improperly exceeded the bounds of the parties' joint pretrial memorandum, (3) that Larkin entered into an impermissible contingent fee arrangement *664for consulting services, and (4) that Larkin's counsel misrepresented the amount of Andrea's past medical bills, causing an "anchoring" effect that inflated the jury award. The judge denied the majority of these motions, but reduced the jury's award in part to adjust for an error in the calculation of Andrea's past medical bills.
Discussion. 1. General Laws c. 231, § 85K. DMA first argues that the judge erred in denying its motion to amend the pleadings to conform to evidence of its charitable status pursuant to Mass.R.Civ.P. 15(a) and (b), 365 Mass. 761 (1974). It also alleges error in the judge's denial of its motion to alter or amend the judgment pursuant to Mass.R.Civ.P. 59(e), 365 Mass. 827 (1974), to conform to the statutory cap on liability as provided in G. L. c. 231, § 85K, as then in effect, which limited the liability of certain charitable organizations to $20,000.8 For substantially the same reasons listed in the judge's thoughtful and comprehensive memorandum and order denying DMA's motions, we disagree.
a. Statutory limitation on liability. The statutory limit set forth in § 85K is an *1217affirmative defense that must be pleaded and proved by the entity seeking to utilize it. See Keene v. Brigham & Women's Hosp., Inc., 439 Mass. 223, 238-239, 786 N.E.2d 824 (2003). "Although technically a limitation on liability, the charitable cap set forth in § 85K has been treated as an affirmative defense that must be pleaded under Mass.R.Civ.P. 8 (c), 365 Mass. 749 (1974) (listing specific affirmative defenses, and concluding with the residuary clause 'any other matter constituting an avoidance or affirmative defense')." Ibid., citing Harlow v. Chin, 405 Mass. 697, 715, 545 N.E.2d 602 (1989). It is undisputed that DMA did not plead the statutory cap as an affirmative defense in its answer. Therefore, as is the case with other affirmative defenses, see Anthony's Pier Four, Inc. v. HBC Assocs., 411 Mass. 451, 471, 583 N.E.2d 806 (1991), DMA waived the statutory limitation on liability.
b. Motion to amend. Rule 15(a) of the Massachusetts Rules of Civil Procedure permits a party to amend the pleadings "by leave *665of court" "when justice so requires." Rule 15(b) allows a party to amend the pleadings to conform to the evidence "[w]hen issues not raised by the pleadings are tried by express or implied consent of the parties[.]" A judge enjoys considerable discretion in deciding whether to allow or deny a motion to amend a complaint. Murphy v. I.S.K. Con. of New England, Inc., 409 Mass. 842, 864, 571 N.E.2d 340 (1991). Here, the judge denied DMA's motion on both rule 15(a) and (b) grounds.
With regard to rule 15(a), as the judge stated in his memorandum and order, DMA "moved to amend its Answer to add the charitable cap as an affirmative defense after more than four years of litigation and two weeks of trial." DMA did not suggest that "this defense was unavailable to it when the Answer was filed in 2011, and, aside from inadvertence, [gave no] reason for the delay." Such a prolonged delay, the judge found, caused "manifest" prejudice to the plaintiffs and made amendment pursuant to rule 15(a) inappropriate. We agree. See DiVenuti v. Reardon, 37 Mass. App. Ct. 73, 77, 637 N.E.2d 234 (1994) ("Among the good reasons ... for which a motion to amend may be denied are that no justification for the lateness of the motion is apparent [beyond counsel for the moving party having had a late dawning idea] and that one or more of the nonmoving parties would be caught off balance by the proffered amendment").9 There was no abuse of discretion.
With regard to DMA's rule 15(b) motion to amend the pleadings to conform to the evidence, the judge found that the parties did not try the issue of DMA's charitable status by either express or implied consent. Although DMA points to its pretrial efforts to have Larkin stipulate to its charitable status, Larkin refused to do so. Furthermore, Larkin objected when, on the last day of trial, DMA sought to introduce its State certificate to prove its charitable status, which was marked for identification to be kept apart from the jury. The plaintiffs objected and reserved their right to argue about DMA's status. The plaintiffs' actions reveal that they did not consent, but rather, that they expressly declined to consent to try the issue of DMA's status. Absent such consent, the judge acted within his discretion to deny DMA's 15(b) motion.
Additionally, DMA claims that the introduction of its State certificate constituted prima facie evidence of its charitable status *666such that the judge erred in denying its motion. We disagree. The documents DMA sought to introduce were marked for identification and were not intended for the jury. "Absent circumstances or an agreement revealing a different approach, *1218... 'documents marked for identification are not in evidence.' " Lingis v. Waisbren, 75 Mass. App. Ct. 464, 470, 914 N.E.2d 976 (2009), quoting from Commonwealth v. O'Neil, 51 Mass. App. Ct. 170, 177 n.7, 744 N.E.2d 86 (2001). Here, the judge did not indicate that he intended to take a different approach, as was taken in Goldberg v. Northeastern Univ., 60 Mass. App. Ct. 707, 711-713, 805 N.E.2d 517 (2004), in which the judge explicitly bifurcated the trial to resolve, separately from the jury, whether the statutory cap applied and, in doing so, marked the university's certificate of incorporation and articles of organization for identification. Instead, here, the judge reassured the plaintiffs that they had reserved their rights as to DMA's (unpleaded) limitation on liability, and indicated that "[i]f it comes to an issue, ultimately that's for post-trial jousting." It was within the judge's discretion to determine that the issue did not arise during the course of the proceedings, making any posttrial "jousting" unnecessary, and to deny DMA's motion.10
2. Larkin's expert witness. DMA next claims that a new trial is warranted because Larkin's expert, Dr. Chavali, testified beyond the anticipated testimony contained within the joint pretrial memorandum. We disagree.
Our rules of civil procedure direct parties to disclose the identity of expert witnesses they plan to call and the "subject matter on which the expert is expected to testify, and to state the substance of the facts and opinions to which the expert is expected to testify and a summary of the grounds for each opinion." Mass.R.Civ.P. 26(b)(4)(A)(i), 365 Mass. 772 (1974). Such disclosures provide notice to the other parties about the intended use of a party's experts, who are considered key factual witnesses in medical malpractice cases. See Kace v. Liang, 472 Mass. 630, 636-640, 36 N.E.3d 1215 (2015). It is within a judge's broad discretion to admit or exclude properly disclosed expert testimony, and, absent prejudicial error, we will not disturb a judge's exercise of that discretion. Id. at 637, 36 N.E.3d 1215.
*667The parties' joint pretrial memorandum indicates that Dr. Chavali was expected to testify that "the size of a venous varix can change substantially and that the rupture rate increases substantially during pregnancy." The parties also anticipated Dr. Chavali would testify that "had a Cesarean Section and/or other alternative treatment ... been performed or offered to Ms. Larkin that to a reasonable degree of medical certainty she would not have suffered an intracranial bleed and the ensuing catastrophic brain injuries." In his trial testimony, Dr. Chavali described his basis for stating that pushing using the Valsalva maneuver was a cause of Andrea's injuries: increased intracranial pressure while pushing prevents blood from exiting, leading to a rupture of the venous varix. Later, he stated that the "venous aneurysm itself may not have ruptured, but [it] was the back-pressure within it from occlusion that caused this rupture. And that rupture is this hemorrhage that's in the brain."
DMA contends that Dr. Chavali testified about causation theories that went beyond the barebones pretrial disclosure. Although the pretrial memorandum did not entail the full range of explanation to *1219which Dr. Chavali testified, what was disclosed was consistent with and not qualitatively different from his trial testimony. In other words, the disclosure was sufficient to provide DMA with notice that the plaintiffs would proffer testimony from Dr. Chavali regarding the nature and causes of the risk of undertaking the Valsalva maneuver in light of the venous varix in Andrea's brain. Dr. Chavali's testimony then provided the jury with a detailed explanation of how such a risk unfolded in this case, namely, the manner in which increased pressure from the Valsalva maneuver built up and caused the venous varix in Andrea's brain to rupture.11 Furthermore, the pretrial memorandum, dated June 2, 2014, supplied DMA with information regarding Dr. Chavali's intended testimony at trial almost one year prior to its commencement, ample time during which DMA could have further explored the matter.
No prejudice resulted, since the pretrial memorandum satisfied the goal of the disclosure rule: "to facilitate the fair exchange of information about critical witnesses and to prevent unfair surprise."
*668Kace v. Liang, 472 Mass. at 636-637, 36 N.E.3d 1215.12 Indeed, DMA's own pretrial expert disclosures demonstrate that DMA was on notice as to Larkin's theory of causation. DMA knew of Dr. Chavali's anticipated testimony and specifically retained multiple experts, including rebuttal testimony that the process of labor did not cause Andrea's hemorrhage. Thus, DMA understood Larkin's causation theory from the disclosure, and the trial judge, who had broad discretion to decide the matter, understood as well. DMA was prepared to and did rebut Dr. Chavali's testimony at trial.
3. Contingency fee arrangement. DMA also claims that it is entitled to a new trial because the judge improperly denied its motion for posttrial discovery regarding the plaintiffs' expert witness compensation. It alleges that the plaintiffs paid a contingency fee to New England Medical Legal Consultants, Inc. (NEMLC), and that such an arrangement undermines the integrity of the judicial process. We disagree.
While payment of a contingency fee to an expert witness is prohibited in Massachusetts,13 see New England Tel. & Tel. Co. v. Board of Assessors of Boston, 392 Mass. 865, 871-872, 468 N.E.2d 263 (1984), no Massachusetts authority has held the same is true of payments made to compensate a consulting service, such as the one retained by the plaintiffs here. Although not binding on us, DMA relies on cases from outside our jurisdiction holding that similar contracts are void as contrary to the established public policy of those States. See, e.g., *1220First Natl. Bank of Springfield v. Malpractice Research, Inc., 179 Ill. 2d 353, 228 Ill.Dec. 202, 688 N.E.2d 1179 (1997). In contrast, however, such contracts have been upheld in other jurisdictions. See, e.g., Schackow v. Medical-Legal Consulting Serv., Inc., 46 Md. App. 179, 197, 416 A.2d 1303 (1980) ("All the experts were to be paid a flat fee by the client. [The consultant]'s role was limited to locating potential experts and then educating them about the case .... That *669arrangement does not violate the public policy of Maryland"). No consistent treatment of consulting contracts emerges from a survey of these extrajurisdictional cases.
The rule that expert witnesses may not collect contingent fees relates to a concern that contingent fees will improperly induce expert witnesses to provide outcome-oriented testimony. See Rule 3.4 of the ABA Model Rules of Professional Conduct (1983) ("A lawyer shall not ... [b] ... offer an inducement to a witness that is prohibited by law"), and comment [3] ("The common law rule in most jurisdictions is that ... it is improper to pay an expert witness a contingent fee"). See also Mass.R.Prof.C. 3.4(b), (g) & comments 3, 5, 426 Mass. 1389 (1998) (adopting American Bar Association model rule). Those same concerns are not directly implicated by the payment of contingency fees to consulting services to locate medical experts, where the medical experts are themselves paid flat fees pro rata based on their time spent preparing for and providing testimony. Where no case or rule (in this jurisdiction) prohibits the practice, we will not upset the judgment on the basis of how these consultants were paid. Moreover, DMA points to nothing in the record that suggests the payment made to NEMLC had any effect on the independence of the expert witnesses who testified for the plaintiffs. Accordingly, we find no abuse of discretion in the judge's denial of DMA's motion for new trial and/or remittitur.
4. Past medical expenses. Finally, DMA claims error in the judge's reduction of the jury award. Although the judge reduced the award for past medical bills upon DMA's motion, DMA now argues that the plaintiffs' misrepresentation of Andrea's past medical bills had an "anchoring" effect -- especially in light of a lack of evidence to support the $11 million award for future medical expenses -- that influenced the entire jury award, which the judge should have vacated. We disagree.
"Questions concerning inadequate or excessive damages are initially within the discretion of the trial judge." Pridgen v. Boston Hous. Authy., 364 Mass. 696, 715, 308 N.E.2d 467 (1974), citing Bartley v. Phillips, 317 Mass. 35, 41-44, 57 N.E.2d 26 (1944). "[A]n award of damages must stand unless ... to permit it to stand was an abuse of discretion on the part of the court below, amounting to an error of law." Reckis v. Johnson & Johnson, 471 Mass. 272, 299, 28 N.E.3d 445 (2015), quoting from Labonte v. Hutchins & Wheeler, 424 Mass. 813, 824, 678 N.E.2d 853 (1997). An error of law occurs "if 'the damages awarded were greatly disproportionate to the injury proven or represented *670a miscarriage of justice.' " Reckis v. Johnson & Johnson, supra, quoting from Labonte v. Hutchins & Wheeler, supra. Furthermore, damages are considered excessive "when they are 'so great ... that it may be reasonably presumed that the jury, in assessing them, did not exercise a sound discretion, but were influenced by passion, partiality, prejudice or corruption.' " Reckis v. Johnson & Johnson, supra, quoting from Bartley v. Phillips, supra at 41, 57 N.E.2d 26.
Here, the misrepresentation, the occurrence of which is not in dispute, arose as follows. Larkin's counsel, in his closing argument, stated that, considering the $4,000 per week Andrea paid for her care, a potential award of $8 million would not *1221"include her four million dollars in medical bills." DMA did not object to this statement at the time it was made. The judge instructed the jury that attorneys' closing arguments do not constitute evidence. After entering their deliberations, the jury returned with a question that indicated the jury may have been influenced by the misrepresentation: "Out of the four million dollars of medical bills, how much was paid out of pocket by the Larkin Family?" In answering the jury's question, the judge instructed the jury that Larkin is "entitled to be ... compensated for those expenses which were reasonable in amount and which were reasonably necessary." To make that determination, the judge explained that "[i]temized medical hospital bills were admitted as evidence of the fair and reasonable charges for such services." Again, neither party corrected the misrepresentation and DMA did not object to it.
The judge determined there was no dispute that, in light of the evidence introduced, Andrea's past medical bills amounted to $1,272,013.70, rather than $4 million, the amount referenced in Larkin's closing argument.14 The award for past medical bills, as the judge found, "likely resulted from mistaken representations to the jury by the plaintiffs at trial and, in any event, ... lacked a sufficient evidentiary foundation to avoid reduction." Accordingly, the judge reduced the award for past medical expenses to the baseline amount reflected in the record.15 DMA does not allege that this reduction was an abuse of discretion.
*671DMA claims that, because the plaintiffs introduced limited evidence as to future medical expenses, the jury may have relied on the misrepresentation to reach its $11 million award for anticipated future medical costs. In arguing a lack of evidence related to future medical costs, DMA cites to (1) Dr. Chavali's statement that Andrea's injuries were permanent; (2) Andrea's father's testimony that her medical costs at the time "roughly" totaled $4,000 per week; and (3) evidence that Andrea's life expectancy was another forty-seven years from the time of trial. DMA claims that this was insufficient evidence on which to base an award for future damages, and, therefore, that the misrepresentation "anchored" the jury, leading to an improper award. We disagree.
Although the plaintiffs did not introduce expert witnesses to prove future medical expenses, DMA points to no case holding that such evidence is required, and it raises no independent argument that the damages award should be vacated on this basis. Cf. Simon v. Solomon, 385 Mass. 91, 105, 431 N.E.2d 556 (1982). Based on the testimony that Andrea's medical care costs "roughly $4,000 a week" and that her injury is permanent -- testimony that the jury were free to credit -- and that her life expectancy is another forty-seven years, the jury could have reasonably calculated that the plaintiffs were entitled to $11 million for future medical expenses. This figure could represent an estimated weekly cost of $4,500,16 totaling $234,000 per *1222year, spread over forty-seven years.17 The jury's reasonable calculation, grounded in the evidence at trial, could have been made without any reference to Larkin's misrepresentation. It is not disproportionate to Andrea's injuries, as it is based in Andrea's current costs, which are likely to continue in the future. Nor does it indicate the jury were "influenced by passion, partiality, prejudice or corruption." Reckis v. Johnson & Johnson, 471 Mass. at 299, 28 N.E.3d 445, quoting from Bartley v. Phillips, 317 Mass. at 41, 57 N.E.2d 26. Therefore, the judge neither erred nor abused his discretion in reducing only the award for past *672medical bills. See Ramos v. Storlazzi, 10 Mass. App. Ct. 876, 877-878, 409 N.E.2d 217 (1980) (finding no abuse of discretion where judge denied motion for new trial based on misrepresentations that were admitted into evidence without objection).
Judgment affirmed.
Orders denying postjudgment motions affirmed.