A MATHEMATICAL MODEL FOR THE SHRINKAGE OF ACOUSTIC NEURINOMA FOLLOWING GAMMA-KNIFE TREATMENT AND SOME RELATIONSHIPS BETWEEN PARAMETERS OF THE FORMULA

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1 A MATHEMATICAL MODEL FOR THE SHRINKAGE OF ACOUSTIC NEURINOMA FOLLOWING GAMMA-KNIFE TREATMENT AND SOME RELATIONSHIPS BETWEEN PARAMETERS OF THE FORMULA AND DOSE-VOLUME HISTOGRAM Hisato NAGANO*1, Hideyo FUJINO*2, Satoshi NAKAYAMA*4, Kazunori TANOHATA*1, Syo MATSUBARA*3 (Received 1 June 1999, accepted 25 January 2000) Abstract: Initial swelling followed by gradual shrinkage is regarded as usual in the Gamma Knife treatment of acoustic neurinomas. We developed a mathematical model representing these changes with 3 independent variables, and a relationship was revealed between these variables and a dose-volume histogram. One hundred and fifty-one patients were treated for acoustic neurinomas between 1992 and 1997 at our facility. Of these, 84 cases could be evaluated with our mathematical model because they were followed by MRI for more than 12 months and at least 5 studies were performed. Peripheral, average and maximum doses were 13.9 }3.1, 18.6 } 4.2 and 28.6 }7.5 Gy, respectively. Pretreatment volumes were 5.5 }9.3 cm3. There were 3 groups, stable (16 cases), rapidly swelling and shrinking (41 cases) and slow-growing and reducible (27 cases). The second group had a small mean time constant of 0.3 months for the initial growing and -23 months for the contiguous shrinkage. The values for the third group were 1 and -43 months, respectively. The degree of swelling in the third group was 3 times higher than that in the second group. The low frequency of the dose distribution over 26 Gy discriminated the members of the second group from the others (p<0.0426), witch indicated that significantly higher doses may not produce good results. Key words: Acoustic neurinoma, Mathematical model, Gamma knife, Dose-volume histogram

2 Formula where R: Residual volumerate Volume at t month Initial volume t : months after GK treatment C : Initial Growth Constant 1/a : Time Constant for late shrinkage 1/b : Time Constant for initial swelling Fig. 1 As mentioned anywhere acoustic neurinomas swelled initially then shrank gradually. The constant b related to the speed of the initial swelling, the degree of which was determined by the constant C. The constant a showed how gradual the shrinkage was.

3 Table 1 Cases were grouped by the values of exp(-b). For exp(-b) see Fig. 1. Several characteristics of the groups were displayed as mean values and doubled standard deviations (2SD). There were significant differences only in the frequency of the dose distribution between groups "indeterminate" and "<0.4" (p<0.006) andbetween "<0.4" and ">0.4" (p<0.043). Fig. 2 Residual rate (R) was plotted against time (month). There are only 2 groups presente here, rapidly swelling and shrinking (exp(-b)<0.4) and slow-growing and reducible (exp(-b)>0.4) groups. Twoexamples of the fitting to formula are shown.

4 永野尚登他 76 うによく一致する(R2=0.95,p= ).残の分布は平均0.001,標準偏差0.107と差なり,予想される体積測定の誤差と矛盾しない. なお,不変群(exp(-b):不 (exp(-b)<0.4),亜定),急性期増大群療前腫瘍体積,周辺,平上の線量区分における分布確率頻度の和を説明因子とする (β=0.25,p<0.0426)が,0.4を急性期増大群(exp(-b)>0.4)の3 群において年齢,治回帰分析に依ればexp(-b)は26Gy以境界値とした場合の2群での判別分析では,この線量域の分布確率頻度の和の大小による正答のパーセントは66%と均,最なった(p<0.0426).な大線量に有為の差は認められなかった(Table1). 時体積,周お,その他の線量域,治なかった,これら急性期増大群,亜の線量容積分布を1Gy毎急性期増大群ステップでの平均値をプロットする事によって比較すると,亜では26Gyよる(Fig.5).な療辺平均最大線量は説明因子にはなら急性増大群り高線領域での頻度が高い事が分かお26Gy以上部分の分布確立頻度の和が説明因子であって,独立2標本のt検定にて両群の平均値間に有為の差を認めるが(0.0284± ,0.105±0.021,p ),個々の線量域の分布頻度に有為差があるわけではない. 考察回帰分析の結果からも本モデルは現象をよく反 Fig. 3 The values of dimensionally of exp(-b) values were compared 0.04 }0.04, significant were plane. larger than 0.4, and smaller to the the contrary respectively). (p<0.0044, plotted C-a (1.0 }0.8) 0.01), and exp(-b) above These had a values group Differences 映しているといえる(Figs2and4).治 three 初の経過観察MRI画 points larger C (0.02 } (0.3 }0.3 像診断が9ヶ月以降となると, 急性期増大群と亜急性期増大群との区別が理論上つけ難くなり,C,aの値が同じ場合,グラフはま were p<0.0000i8). Fig. 5 Fig. 4 療後の最 The graph shows a good correlation between the observed residual rate (obsr) and calculated residual rate (calc R). The values of calcr were estimated from the formula in Fig. 1. This is a graph of the average values of the differential dose histogram of the two groups, rapidly swelling and shrinking (exp(-b)<0.4) and slow-growing and reducible (exp(-b)>0.4) groups. There was no significant difference between the two values of each dose class, however, summing up frequencies of classes over 26 Gy, a good discriminant between the 2 groups was derived.

1) Noren G: Gamma Knife Radiosurgery for Acoustic Neurinomas, Textbook of Stereotactic and Functional Neurosyrgery, Philip L. Gildenberg, McGraw-Hill, 1997, pp 835-844.

5 1) Noren G: Gamma Knife Radiosurgery for Acoustic Neurinomas, Textbook of Stereotactic and Functional Neurosyrgery, Philip L. Gildenberg, McGraw-Hill, 1997, pp ) Flickinger JC, Lunsford LD, Coffey RJ, et al: Radiosurgery of acoustic neurinomas. Cancer 67 (2): , ) Flickinger JC, Kondziolka D, Lunsford LD: Dose and cliameter relationships for facial, trigeminal, and acoustic neuropathies following acoustic neuroma radiosurgery. Radiother Oncol 41: 215-9, ) Ogunrinde OK, Lunsford LD, Flickinger JC, et al: Stereotactic radiosurgery for acoustic nerve tumors in patients with useful preoperative hearing: results at 2- year follow-up examination. J Neurosurg 80 (6): , ) Kondziolka D, Lunsford LD, McLaughlin MR, et al: Long-term outcomes after radiosurgery for acoustic neuromas. N Engl J Med 339 (20): , 1998.

6) Nora G: Long-term complications following gamma knife radiosurgery of vestibular schwannomas. Stereotact Funct Neurosurg 70 Suppl 1: 65-73, 1998.

6 6) Nora G: Long-term complications following gamma knife radiosurgery of vestibular schwannomas. Stereotact Funct Neurosurg 70 Suppl 1: 65-73, ) Pollock BE, Lunsford LD, Noren G: Vestibular schwannoma management in the next century: a radiosurgical perspective. Neurosurgery 43 (3): ; discussion 481-3, ) To SY, Lufkin RB, Rand R, et al: Volume growth rate of acoustic neuromas on MRI post-stereotactic radiosurgery. Comput Med Imaging Graph 14 (1): 53-9, ) Linskey ME, Lunsford LD, Flickinger JC: Neuroimaging of acoustic nerve sheath tumors after stereotaxic radiosurgery. AJNR Am J Neuroradiol 12 (6): , ) Mendenhall WM, Friedman WA, Bova FJ: Linear accelerator-based stereotactic radiosurgery for acoustic schwannomas. Int J Radiat Oncol Biol Phys 28 (4): , ) Yamamoto M, Jimbo M, Ide M, et al: Is unchanged tumor volume after radiosurgery a measure of outcome? Stereotact. Funct Neurosurg 66 Suppl 1: 231-9, ) Flickinger JC, Lunsford LD, Linskey ME, et al: Gamma knife radiosurgery for acoustic tumors: multivariate analysis of four year results. Radiother Oncol 27 (2): 91-8, ) Nagano H, Tanohata K, Kato E, et al: Dose distribution and shrinkage of acoustic neurinomas 2 years after Gamma Knife treatment. Stereotact Funct Neurosurg 66 Suppl 1: , ) Fujino H, Inomori S, Nakayama S, et al: Gamma Knife Radiosurgery for Vestibular Schwabbomas. Stereotactic Radiosurgery 3: 35-45, ) Flickinger JC, Kondziolka D, Lunsford LD: Dose and diameter relationships for facial, trigeminal, and acoustic neuropathies following acoustic neuroma radiosurgery. Radiother Oncol 41 (3): 215-9, ) Linskey ME, Martinez AJ, Kondziolka D, et al: The radiobiology of human acoustic schwannoma xenografts after stereotactic radiosurgery evaluated in the subrenal capsule of athymic mice. J Neurosurg 78 (4): , ) Hirato M, Inoue H, Zama A, et al: Gamma Knife radiosurgery for acoustic schwannoma: effects of low radiation dose and functional prognosis. Stereotact Funct Neurosurg 66 Suppl 1: , ) S Fukuoka, K Oka, Y Seo, et al: Apoptosis following Gamma Knife Radiosurgery in a Case of Acoustic Schwannoma. Stereotactic and Functional Neurosurgery 70 S 1: ) Linskey ME, Lunsford LD, Flickinger JC, et al: Stereotactic radiosurgery for acoustic tumors. Neurosurg Clin N Am 3 (1): , ) Seo Y, Fukuoka 5, Nakagawara J, et al: Effect of Gamma Knife radiosurgery on acoustic neurinomas. Assessment by 99mTc-DTPA-human serum albuminand 201 T1C1-single photon emission computed tomography. Stereotact Funct Neurosurg 66 Suppl 1: , ) Hirato M, Hirato J, Zama A, et al: Radiobiological effects of gamma knife radiosurgery on brain tumors studied in autopsy and surgical specimens. Stereotact Funct Neurosurg 66 Suppl 1: 4-16, ) Andrews DW, Silverman CL, Glass J, et al: Preservation of cranial nerve function after treatment of acoustic neurinomas with fractionated stereotactic radiotherapy. Preliminary observations in 26 patients. Stereotact Funct Neurosurg 64 (4): , 1995.

Studies of Foot Form for Footwear Design (Part 9) : Characteristics of the Foot Form of Young and Elder Women Based on their Sizes of Ball Joint Girth

Studies of Foot Form for Footwear Design (Part 9) : Characteristics of the Foot Form of Young and Elder Women Based on their Sizes of Ball Joint Girth and Foot Breadth Akiko Yamamoto Fukuoka Women's University,