Application of Invasion Mathematical Model in Dosimetry for Boron Neutron Capture Therapy for Malignant Glioma K. Yamamotoa,*, H. Kumadaa,b, K. Nakaib, K. Endob, T. Yamamotoc, A. Matsumurab a
Japan Atomic Energy Research Institute, Shirakata-shirane 2-4, Tokai-mura, Ibaraki, 319-1195, Japan b
c
University of Tsukuba, Tenoudai 1-1-1 , Tsukuba, Ibaraki, 305-8577, Japan
Ibaraki Prefectural Central Hospital, Koibuchi 6528, Tomobe, Ibaraki, 309-1793, Japan
*Corresponding author.
Tel .:+81-29-282-5746; Fax.: +81-29-282-5258. E-mail address:
E-mail address:
[email protected] (K. Yamamoto)
Abstract A dose distribution considered the tumor cell density distribution is required on the general principle of radiation therapy.
We propose a novel method of determining target
region considering the tumor cell concentration as a new function for the next generation Boron Neutron Capture Therapy (BNCT) dosimetry system. It has not been able to sufficiently define the degree of microscopic diffuse invasion of the tumor cells peripheral to a tumor bulk in malignant glioma using current medical imaging.
Referring to treatment
protocol of BNCT, the target region surrounding the tumor bulk has been set as the region which expands at the optional distance with usual 2cm margin from the region enhanced on T1 weighted gadolinium Magnetic Resonance Imaging (MRI). In this research, the cell concentration of the region boundary of the target was discussed by using tumor cell diffusion model in the sphere spatio-temporal system. The survival tumor cell density distribution after the BNCT irradiation was predicted by the two regions diffusion model for
a virtual brain phantom. Keywords: Boron Neutron Capture Therapy, Diffusion model, Invasion, Brain Tumor, Malignant Glioma
1.
Introduction
The malignant glioma is characterized by their aggressive diffuse invasion to the surrounding normal tissue.
The development of medical imaging has improved the
detectability of gliomas but has not been able to sufficiently define the degree of diffuse invasion of the tumor cells peripheral to the bulk of the tumor mass to allow for adequate treatment planning.
Thus, it is not surprising that even extensive surgical excision or
radiation therapy with local high intensity irradiation is followed by recurrence at or near the edge of the excision (Gaspar et al., 1992).
Recently, several researchers tried predictions in
the survival time, etc. by the application of the mathematical model on the tumor cell invasion process, which introduced the infiltration characteristic from the research field of physic and mathematic.
In this research, we calculated the cell density of the region
boundary of the target using tumor cell diffusion model of Burgess et al (1997). Further, survival tumor cell density distribution after the BNCT irradiation will be predicted by two matters region diffusion model of Swanson et al. (2002). Based on these studies, a new concept for BNCT dosimetry system is proposed.
2.
Material and method
A simple model uses the differential equation for the total number of tumor cells. This mathematical model ignores the spatial aspects of the disease. Let nt be the number of tumor
cells at time t. Then the model states that the time derivative of the total tumor, is given by dn t = pn t − knt , dt
(1)
where p is the proliferation rate of tumor and k is the killing rate of tumor. We define flowing as a net proliferation ρ.): ρ = p−k .
(2)
The net proliferation rate ρ is estimated 0.012day-1 from the doubling time, which is about two months for malignant glioma (Swanson 2002, Lawrence et al. 2002, Tracqui et al. 1995).
2.1 Spherically Symmetric Model
A simple spatio-temporal model was represented as diffusion equation. In the spherically symmetric model of Burgess et al. (1997), they assumed that the tumor cells diffuse freely in the brain and proliferate exponentially as the following: ∂n 1 ∂ 2 ∂n =D 2 r + ρn , ∂t r ∂r ∂r
(3)
where n is the concentration of tumor cells at location r at time t, and D is the diffusion coefficient in this partial differential equation. If the tumor starts from a point source of initial cells concentration N0 at time t0, the solution to this diffusion equation (3) at time t is: n( r , t ) =
N0
(4πDt )3 / 2
r2 . exp ρt − 4 Dt
(4)
Swanson et al. (2002) assume that the initial tumor cells concentration is about 4000cells as local mass before diffusing. Burgess et al. (1997) assumed that a tumor is diagnosed when the tumor reaches visually detectable diameter, which is defined as having a cell concentration Nc. Above the threshold value of 8x106 cells/cm3, the tumor diameter reaches 3cm and that a patient dies when this diameter reaches 6cm. The remained cell distribution is calculated by using 9L gliosarcoma survival fraction
curve (SF=exp(-0.9Dose_BN)) which had been obtained from the cell irradiation examination of JRR-4 beam performance (Yamamoto et al., 2001).
Dose_BN is absorbed doses of
boron-neutron reaction and nitrogen-neutron reaction for a thermal neutron point source, which place in the tumor center.
2.2 Realistic Digital Brain Phantom
Realistic digital brain phantom model of Swanson et al. (2002) define different diffusion coefficients of white and gray matter because the transfer speed of the invasion cells in both components are different. Swanson mathematical model for glioma growth and tumor invasion can be written as: ∂n = ∇ ⋅ (Di ∇n ) + ρn , ∂t
(i=g or w)
(5)
where n is the concentration of tumor cells at location and time t. The diffusion coefficient Di is selected a value of which Dg or Dw for gray or white matter at the location, respectively. The diffusion coefficients ratio of white matter for gray matter was estimated 5 times by Swanson et al. (2002). To complete the model formulation, they impose a ‘zero-flux’ of cells across the brain boundaries defined by cerebrospinal fluid in the ventricular and subarachnoid space. This boundary condition requires that glioma cells are not allowed to migrate outside of the brain tissue. Diffusion constant Dw of the white matter is 0.0024 cm2/day and diffusion constant Dg of the gray matter is 0.00048 cm2/day (Swanson 2002). The equation (5) was differentiated according to the Crank-Nicholson scheme and was encoded by the algorithm using the SOR method (Successive Over-Relaxation method) in the implicit step. We also simulate tumor cells motility with using the Realistic Digital Brain Phantom of Collins et al. (1998). The digital data of the phantom has been published on home page as http://www.bic.mni.mcgill.ca/brainweb/. In order to simulate the glioma on the right frontal lobe, the initial tumor source with density of 4000 cells/mm3 was put into the rectangular
coordinates (71, 144, 95) on the phantom data.
3.
Results
3.1 Cell Concentration on Target Boundary
Figure 1 shows the spatial distribution of tumor cell concentration. The broken line shows the condition before the BNCT irradiation, which takes on the visually detectable tumor average diameter (enhancedarea on T1weighted gadolinium MRI) of 3cm. The solid line shows the condition after the BNCT irradiation, which shows the distribution of the visually undetectable cell. The target volume selected BNCT include 2cm margin from the tumor bulk. The target boundary cells concentration at 2cm margin is calculated from equation (4) so that this value is 1.22x106 cells/cm3. The distribution of post-irradiation has one peak of the cell concentration at 5.6cm radius.
3.2 Simulation of Tumor Cell Concentration on Fontal Lobe Glioma in Distal Brain
Figure 2 shows the simulated distribution on frontal lobe glioma cells in the digital brain. The dark gray area indicates the tumor bulk region with the surrounding boundary that is defined as having a cell concentration above the threshold value of 8x106 cells/cm3. The light gray area indicates the target that defined as having a cell concentration above the threshold value of 1.22x106 cells/cm3. This result means a high possibility of new setting method of the target region including the undetectable tumor by using the analytical model that corresponds to the cell concentration distribution.
4.
Discussions
The current stream of mathematical research will provide great insights into important
problem if the mathematical analysis will be able to combine with laboratory simulation and clinical treatment planning system. Especially, the dynamic spatial model of Swanson et al. (2002) is effective for developing the dose planning system. This model is intensively introduced in the present research.
4.1 Simulation of Recurrence
When mathematical model by Swanson et al. (2002), will be combined with dose planning system such as JCDS (Kumada and Torii, 2002, Kumada et al., 2002) or SERA (Wheeler et al., 1999), we may select the optimum target region including the undetectable tumor. The calculation for re-growth of residual tumor in the phantom after a thermal neutron BNCT treatment was performed. The calculation showed, that after 22.4 months from the irradiation, the tumor recurrence became visible. Figure 3 shows the recurrence in the left-brain. This result means that 1) the remaining tumor cells spread through the corpus callosum, 2) they microscopically invade the full brain and 3) they colonized in the area where is far away from the treated area, as also shown from the result of Figure 1. Thus combination with dose planning system and cell diffusion model will provide the medical and biological information for predicting on the recurrence of a malignant glioma.
4.2 Survival time
The death in this simulation occurs when the tumor volume reach a size equivalent to a sphere with an average 6cm diameter. In case of this digital phantom, the survival times were 2.9 months and 23.7 months with and without the BNCT treatment, respectively. When the patient become symptomatic with the recurrence, his life expectancy becomes 1.3 months. At this moment, the retreatment is expected to provide 2.7 months in life expectancy that is shorter than the effect of initial treatment. It is noted that cell density will be lower in the
recurrence than in the initial stage, when comparing the two pictures which indicated the cell density distributions (Figure 3). The latter has the wide and low cell distribution. The tumor cells invade under the detectable limit for a certain period and then it become suddenly visible when it reaches a certain size. This may be misinterpreted as a rapid tumor re-growth. Thus the treatment plans may be used to estimate the survival time quantitatively.
4.3 Detectable Tumor Volume Growth
In above, we discussed that the tumor diameter rapidly was increased. Total number of tumor cells has been grown exponentially because the net proliferation rate is estimated as a constant coefficient. But the detectable tumor volume has not been expanded exponentially. The equation (4) is rearranged at the radius and is differentiated in the time t, and the equation of the radius growth rate is deduced as following:
∂rc = ∂t
2 Dρt 2 − 3D
n 2/3 2t Dρt − Dt ln c (4πDt ) N0
,
(6)
2
where nc is the threshold tumor concentration of 8x106 cells/cm3 on the boundary with tumor bulk region and surrounding infiltration region, and is a limiting line that can visualize tumor by enhanced CT imaging. Therefore, the detectable volume growth rate is calculated as:
n ∂Vc ∂Vc ∂rc 8π 2/3 2 Dρt 2 − 3D Dρt 2 − Dt ln c (4πDt ) . = = t ∂t ∂rc ∂t N0
(
)
(7)
Figure 4 shows obviously the different growth curves for detectable volume growth and total tumor cell proliferation.
4.4 Problem of Application for Targeting
It has to be reminded that the initial purpose of this method is to define the target region considering the tumor cell concentration for the future dose planning system. This paper explains only the cell concentration on the boundary. In order to select target region, we should know the initial distribution of the tumor cell from the medical images. In the future, by using this model much information could be provided for medical and biological data. This could be linked to the dose planning system. By using this model, establishment of an intelligent treatment planning system for radiation therapy including BNCT may become feasible.
Acknowledgements
We are thankful to the members of the Center for Promotion of Computational Science and Engineering at JAERI for helpful discussions of computing algorisms.
References
Burgess P.K., Kulesa P.M., Murray J.D., Alvord Jr, E.C., 1997. The interaction of growth rates and diffusion coefficients in a three-dimensional mathematical model of gliomas. J Neuropathol. Exp. Neurol. 56, 704-713. Collins, D.L., Zijdenbos, A.P., Kollokian, V., Sled, J.G., Kabani, N.J., Holmes, C.J., Evans, A.C., 1998. Design and contruction of a realistic digital brain phantom. IEEE Trans. Med. Imag. 17, 463-468. Gaspar, L.E., Fisher, B.J., Macdonald, D.R., LeBer, D.L., Halperin, E. C., Schold, S. C.,
Cairncross, J.G., 1992. Supratentorial malignant gliomas: patterns of recurrence and implications for external beam local treatment. Int. J. Radiat. Oncol. Biol. Phys. 24, 55-57. Kumada, H. and Torii, Y., User’s manual of a supporting system for treatment planning in Boron Neutron Capture Therapy - JAERI Computational Dosimetry System -. 2002. JAERI-Data/Code 2002-018 (in Japanese). Kumada, H., Matsumura, A. Nakagawa, Y., Yamamoto, K., Yamamoto, T., 2002. Verification of the JAERI Computational Dosimetry System for Neutron Capture Therapy. In: Sauerwein, W., Moss, R., Wittig, A. (Eds.), Research and Development in Neutron Capture Therapy. Monduzzi Editore, Bologna Italy, pp. 529-534. Wein, L.M., Joseph, T.W., Alexandra, G. I., Raj K.P., 2002. A mathematical model of the impact of infused targeted cytotoxic agents on brain tumours: implications for detection, design and delivery. Cell Prolif. 35, 343-361. Swanson K. R., Alvord Jr, E. C., Murray J. D., 2002. Virtual brain tumours (gliomas) enhance the reality of medical imaging and highlight inadequacies of current therapy. British J. Cancer 86, 14-18. Tracqui, P., Cruywagen, G. C., Woodward, D. E., Bartoo, G. T., Murray, J. D., Alvord Jr, E. C., 1995. A mathematical model of glioma growth: the effect of chemotherapy on spatio-temporal growth. Cell Prolif. 28(1), 17-31. Yamamoto, K., Yamamoto, T., Kumada, H., Torii, Y., Kishi, T., Matsumura, A., Nose, T., Horiguchi, Y., 2001. Evaluation of JRR-4 Neutron Beam using Tumor Cells. JAERI-Tech 2001-017 (in Japanese). Wheeler, F. J., Wessol, D. E., Wemple, C. A., Albright, C. L., Cohen, M.T., Frandsem, M. W., Harkin, G. J., Rossmeier M.B., Nigg, D. W., 1999. SERA - An advanced treatment planning system for neutron therapy. INEEL/CON-99-0523.
The illustration Captions
Figure 1. The spatial distribution of tumor cell concentration: The broken line shows the condition before the BNCT irradiation, which is that the visually detectable tumor average diameter is 3cm. The solid line shows the condition after the BNCT irradiation, which shows the distribution of the visually undetectable cell.
Figure 2. The simulation on fontal lobe glioma cells in the distal brain. The dark gray area indicates the tumor bulk region surrounding with the frontier that is defined as having a cell concentration above the threshold value of 8x106 cells/cm3, the light gray area indicates the target that defined as having a cell concentration above the threshold value of 1.22x106 cells/cm3.
Figure 3. Simulation after the BNCT irradiation. a) before irradiation, b) the absorbed dose distribution of boron - neutron reaction plus nitrogen - neutron reaction. c) After 22.4 months from the irradiation, the tumor became recurrence.
Figure 4. The growths curve of detectable tumor volume and total tumor cells. The growth curves for detectable volume growth and total tumor cell proliferation are different.
1.0E+07 8.0E+06 6.0E+06 4.0E+06 2.0E+06
Target with 2cm margin
1.2E+07
Tumor Bulk 3cm
Tumor Cells Concentration, n （cells/cm3）
1.4E+07
7.0E+02
Before the BNCT Irradiation After the BNCT Irradiation
6.0E+02 5.0E+02 4.0E+02 3.0E+02 2.0E+02 1.0E+02
0.0E+00
0.0E+00 0
1
2 3 4 5 6 7 8 9 10 Radial Distance from Tumor Center, r (cm)
11
Figure 1. The spatial distribution of tumor cell concentration: The broken line shows the condition before the BNCT irradiation, which is that the visually detectable tumor average diameter is 3cm. The solid line shows the condition after the BNCT irradiation, which shows the distribution of the visually undetectable cell.
Target
Tumor bulk
Figure 2. The simulation on fontal lobe glioma cells in the distal brain. The dark gray area indicates the tumor bulk region surrounding with the frontier that is defined as having a cell concentration above the threshold value of 8x106 cells/cm3, the light gray area indicates the target that defined as having a cell concentration above the threshold value of 1.22x106 cells/cm3.
1.0E+03
1.0E+09
1.0E+02
1.0E+08
1.0E+01
1.0E+07
1.0E+00 Total number of cells
1.0E+06
Detactable Tumor Volume
1.0E-01
Initial Dignosis 1.0E+05 1.0E+04 980
1.0E-02
Deth
1030
1080 Time t (days)
1130
1.0E-03 1180
Figure 4. The growths curve of detectable tumor volume and total tumor cells. The growth curves for detectable volume growth and total tumor cell proliferation are different.
3
1.0E+10
Detectable Tumor Volume Vc (cm )
Tumor cells n (cells)
Figure 3. Simulation after the BNCT irradiation. a) before irradiation, b) the absorbed dose distribution of boron - neutron reaction plus nitrogen - neutron reaction. c) After 22.4 months from the irradiation, the tumor became recurrence.
(500words abstract)
Application of Invasion Mathematical Model In topic #04: Physics Authors K. Yamamoto, Japan Atomic Energy Research Institute H. Kumada, Japan Atomic Energy Research Institute K. Nakai, University of Tsukuba K. Endo, University of Tsukuba T. Yamamoto, Ibaraki Prefectural Central Hospital A. Matsumura, University of Tsukuba Summary:
A dose distribution considered the tumor cell density distribution is required on the general principle of radiation therapy. We propose a novel method of determining target region considering the tumor cell concentration as a new function for the next generation Boron Neutron Capture Therapy (BNCT) dosimetry system. It has not been able to sufficiently define the degree of microscopic diffuse invasion of the tumor cells peripheral to a tumor bulk in malignant glioma using current medical imaging. Referring to treatment protocol of BNCT, the target region surrounding the tumor bulk has been set as the region which expands at the optional distance with usual 2cm margin from the region enhanced on T1 weighted gadolinium Magnetic Resonance Imaging (MRI). The malignant glioma is characterized by their aggressive diffuse invasion to the surrounding normal tissue. Recently, several researchers tried predictions in the survival time, etc. by the application of the mathematical model on the tumor cell invasion process, which introduced the infiltration characteristic from the research field of physic and mathematic. The current stream of mathematical
research will provide great insights into important problem if the mathematical analysis will be able to combine with laboratory simulation and clinical treatment planning system. Especially, the dynamic spatial diffusion model is effective for developing the dose planning system. In this research, we calculated the cell density of the region boundary of the target using tumor cell diffusion model. Further, survival tumor cell density distribution after the BNCT irradiation will be predicted by two matters region diffusion model. Based on these studies, a new concept for BNCT dosimetry system is proposed. The target boundary cells concentration at 2cm margin is calculated from the diffusion model so that this value is 1.22x10^6 cells/cm^3. The distribution of post-irradiation has one peak of the cell concentration at 5.6cm radius. The distribution on frontal lobe glioma cells in the digital brain simulated. This result means a high possibility of new selecting method of the target region including the undetectable tumor by using the analytical model that corresponds to the cell concentration distribution. In order to select the target region, we should know the initial distribution of the tumor cell from the medical images. In the future, much information for medical and biological data, which include the survival tumor cell distribution, the survival time, the date of recurrence, the location of recurrence, the information for combined other therapy after that and etc., could be provided, when the diffusion model could be linked to the dose planning system. By using this model, establishment of an intelligent treatment planning system for radiation therapy including BNCT may become feasible.
