In1 .I Rndiatmn Oncologr BioL Phyc, Vol. Printed in the U.S.A. All rights reserved.
19. pp.
0360.3016/90 $3.00 + .Xl Copyright C 1990 Pergamon Press plc
203-207
0 Oncology Intelligence THE BIOLOGICAL EFFECT OF INHOMOGENEOUS DOSE DISTRIBUTIONS IN FRACTIONATED RADIOTHERAPY ROBERT J. YAES, Sc.D., M.D. Department
of Radiation Medicine, University of Kentucky Medical Center, Lexington, KY 40536-0084 U.S.A.
The linear quadratic (LQ) model is applied to an organ receiving a fractionated course of radiotherapy with an inhomogeneous dose distribution. It is shown that the gradient in the extrapolated response dose (ERD) will be steeper than the gradient in the physical dose. This effect will be greatest for an organ with a small a/@ ratio treated with large dose fractions. Clinical implications are discussed with an emphasis on radiation myelitis. Radiation myelitis, Linear quadratic model, Late effects, Normal tissue damage.
effect, since both the fraction size and the total dose will be higher. This double effect can be easily quantified by using the linear-quadratic (LQ) model. Since, in the LQ model, the response to changes in fraction size will depend on the a/p ratio (2, 6, 17, 24, 26) this response will vary from organ to organ. For simplicity, the effect of an inhomogeneous dose delivered to a single organ is assessed. In an organ we can choose a point at or near the center of the organ as the origin and take the dose at this point D (a) given in N fractions of size d (6) to be the “average” dose received by the organ. Of course
INTRODUCTION In an animal experiment ( 1,6,9, 18, 19,2 1) whose object is to determine normal tissue tolerance, a uniform dose of radiation is delivered to an entire organ or to a specified volume of tissue. Total doses, fraction sizes and treatment volumes can be varied in a systematic way, using a large number of individual animals, so that detailed dose-response curves can be obtained. Models such as the linearquadratic (LQ) model (2,6) can be tested and the parameters of these models can be accurately determined. It is not clear however, how normal tissue tolerance data, obtained from experimental animals can be extrapolated to man (1). In a clinical situation, our objective is not to deliberately induce normal tissue damage, but, on the contrary, to attempt to cure a patient’s cancer while minimizing the liklihood of clinically significant complications. Although it is intended that a uniform dose be delivered to the entire treatment volume, when large fields must be used to include all areas of potential microscopic tumor extension, inhomogeneities invariably occur. As a result, for each daily treatment, small volumes of normal tissue may receive doses that are consistently higher than the prescribed dose. Goiten (8) has considered the effect of inhomogeneties on the liklihood of tumor control so we will limit our discussion to the problem of normal tissue damage. METHODS
AND
D (5) = N d (-6).
Any other point in the organ can be specified by the vector 51= (x, y, z) and the dose at the point 51is D (si) = Nd(SI)
(2)
where d (2) is the fraction size received at point 51(Fig. 1). It is explicitly assumed that the same isodose distribution is valid for each fraction. If the variation in dose and fraction size within the organ is small, we can write D (2) = D (5) + AD(Z) d (X) = d(Ti) + Ad (X)
MATERIALS
AD(Z) = NAd(St).
Application of the linear-quadratic model to an inhomogeneous dose distribution If a volume of normal tissue receives a daily dose that is higher than the prescribed dose, there will be a double
Accepted for publication
(1)
(3)
We assume that Ad(Z) is small compared to d(a) and thus AD@) is small compared to D(a).
24 January 1990. 203
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July 1990, Volume 19, Number 1
Since D(Z) = Nd(Z), Eq. (8) becomes D,(Z)
Fig. 1. Non-uniform dose distribution. The origin o within an organ receives a dose D in N equal fractions of size d. Any other point within the organ can be specified by the vector 2 = (x, y, z). The dose at the point TIis D(Z) given in N equal fractions of size d(Z). We explicitly assume that the same isodose distribution is valid for each fraction.
In the linear quadratic model, organ damage is assumed to result from the depletion of a target stem cell population (6, 15) whose survival probability, S has the form S = eeE,
E = cuD + @D2.
(4)
If the total dose D is given in N equal fractions of size d, and if sufficient time is allowed for complete repair of sublethal damage between fractions, instead of Eq. (4) we would have (6, 17) E = aD + /3Dd.
(5)
It is also an assumption of the LQ model that “isoeffect equals isosurvival,” that is, volumes of tissue that have the same value of E and hence the same value of S will have the same level of damage (6). It is easy to show that the equation for the isoeffect curves in the d-D plane (the values of d and D that produce the same value of E and hence of S) has the form (6, 17, 24, 26) D(d + 6) = 6D,
(6)
D,
(7)
or = D(d + 6)/6
Response Dose, where D, = E/a is the Extrapolated (ERD) and 6 = CX/~is the (Y/P ratio and has the dimensions of dose. We wish to write Eq. (7) for the origin Ti and the point x’. For simplicity, we will define D = D(a), D, = D,(G), d = d(a). Then the equation at 6 will be identical to Eq. 7. At x we will have
D,(g)
= D(Sl)(d(ST) + 6)/a.
(8)
= D(Z) + D’(?)/NS.
(9)
D(Z) is the spatial dose distribution. The value of D(Z) at point Z is the physical dose that would be measured by a small ionization chamber placed at the point TI. D,(Z) is a measure of the cell survival probability, S, at the point Z and can therefore be thought of as the “biologically effective” dose distribution. D,(Z) will depend on the fraction size and on the a//3 ratio, as well as on the physical dose distribution D(Z). Curves along which D(Z) remains constant are the isodose curves. From Eq. 9, it is clear that D,(X) will also be constant along such curves. Thus, as one might expect, isodose curves are also isoefict curves. However, the gradient in biologically effective dose will be greater than the gradient in physical dose as we can see by differentiating Eq. 9. + D,(x)
= [(2d t 6)/6] - + D(x).
(10)
If we again assume that the variation in dose over the treatment field is small, then, for the change in dose as we go from the origin 6 to the point 51we would have A D(Z) = 2-G A D,(Z)
= Sr.?
It is then easy to see by dividing using Eqs. 8, 10 and 11,
D(SI)
(11)
D,(Z).
(12)
Eq. 12 by D,(X)
and
A D, (3/D, C-4 = [(2d + 6)/(d + S)] - [A D(s;)/D(s;)].
(13)
The significance of Eq. 13 can best be illustrated with some simple examples. Consider a late reacting tissue with an a/p ratio of 2 Gy treated with 2 Gy fractions. The proportionality factor, (2d + 6)/(d + a), is then 3 = 1.5. Thus, in this case, if the dose D(Z) is 10% higher at ? than at i3, the biologically effective dose at this point would be 15% higher. If the same organ were treated with a hyperfractionated regimen with 1 Gy fractions, a 10% increase in physical dose would lead to a 13% increase in biologically effective dose whereas for a hypofractionated regimen with 5 Gy fractions, the same 10% increase in physical dose would give a 17% increase in biologically effective dose. For an early reacting tissue, with an (Y/P ratio of 10 Gy treated with 2 Gy fractions, a 10% increase in physical dose would result in less than a 12% increase in biologically effective dose. Eq. 13 implies that if there is a volume of normal tissue that consistently receives a slightly higher dose than the prescribed dose, the effect on this tissue will be magnified by a factor of (2d + 6)/(d + 6). This magnification factor
Biological effect of inhomogeneous dose distributions 0 R. J. YAES
will be greatest for an organ with a small CY//~ratio treated with large dose fractions. Thus, there is a second beneficial effect of hyperfractionation in addition to the direct sparing of late reacting tissues ( 17) with small (Y/Pratios. The effect of any “hot spots” involving these same organs will be minimized. The extrapolated response dose, D,(Z), is proportional to the logarhythm of the survival probability S for the target stem cell population (2). The relationship between the survival probability, S and the probability, P, of organ damage, however, is complex and depends on the details of the functional anatomy of the organ in question. A useful concept is the analogy with electrical circuits (25) whereby many organs can be considered to be either “series organs” or “parallel organs.” A “parallel organ” consists of a large number of structurally and functionally independent subunits all performing the same function “in parallel”, so that the output of the organ as a whole is the sum of the outputs of the individual subunits. If the organ is damaged, the degree of damage will be proportional to the number of subunits that have been inactivated. Examples of parallel organs would be the kidney composed of nephrons or the lung composed of alveoli. In a series organ, tissue continuity is essential for organ function. Disturbance of that continuity by destruction of even a very small volume of tissue could cause the organ to cease functioning entirely. An example of a series organ would be the spinal cord (23, 27, 28). A model that would relate the probability of organ damage to the cell survival probability has been discussed in detail elsewhere (10, 22, 23, 27, 28). There is no need to repeat that discussion here. For the purpose of this discussion a simple approximation might suffice. For organs with steep dose-response curves (7) we can assume that for each type of tissue there is an extrapolated tolerance dose, (2) D’, . When the extrapolated response dose D,(Z) exceeds the extrapolated tolerance dose D’, in a given volume of tissue, the tissue within that volume will be rendered non-functional. Hence if we can determine the spatial distribution of the extrapolated response dose within an organ, D,(Z), and if we know the extrapolated tolerance dose D’, for that organ, we should be able to predict how much damage that organ would suffer. Computerized treatment planning systems are now available that can use digitalized CAT scan data to determine the dose distribution in 3-dimensional space within any organ ( 13). If the (Y/Pratio for the organ is known and the fraction size is specified, it would be a simple matter to modify these systems so as to calculate the biologically effective dose distribution as well. RESULTS Radiation myelitis
The organ for which these considerations will be most important will be the spinal cord for several reasons.
205
Firstly and probably most important, the spinal cord (1, 3, 4, 5, 9, lo, 11, 12, 14, 16, 18, 19, 20, 21, 22, 27, 28) can be considered all to be a “series organ.” Destruction of a very small volume if tissue will lead to a devastating complication (22,25,27,28). Destruction of a thin “slice” of spinal cord tissue, by radiation or by any other means, can result in cord transection, complete loss of all neurological function below the level of the lesion. Thus, an inhomogeneity or overlap that causes the overdosing of a very small volume of spinal cord tissue can have clinically significant results. Secondly, the a/P ratio for spinal cord of between 1 and 2 Gy is small even for late reacting tissue (1). Thus, for a given fraction size the magnification factor of (2d + 6)/(d + 6) will be larger for the spinal cord than for most other organs. It is for this reason that overlap on the spinal cord between adjacent fields that are treated simultaneously could be more dangerous than a similar overlap between fields that are treated sequentially (retreatment). When there is overlap between fields that are treated simultaneously, both the fraction size and the total dose will be increased in the area of overlap. When there is overlap between fields that are treated sequentially, there will be an increase in the total dose but not in the fraction size in the area of overlap. (There are, however both experimental and theoretical reasons to believe that very small areas of overlap, less than 0.5 cm, could be tolerated (9, 10, 18, 22, 27, 28)). Thirdly, the dose response curve for radiation myelitis is very steep, (7, 19, 2 1) so that even a small change in dose can result in a large change in the complication probability. Thus, when the single dose of radiation received by the rat lumbar spinal cord (19) is increased by 5% from 1900 to 2000 cGy, the probability of myelitis increases from 0% to 100%. Even when the dose to the lumbar cord (19) is given in 10 equal fractions, a 20% increase in dose from 4500 to 5500 cGy can also increase the complication probability from 0% to 100%. Although detailed dose-response curves for the human spinal cord are not available, it is possible that they could be comparably steep. Fourthly, the shape of the spinal cord is long and thin. There is likely to be a greater variation in dose over the length of the spinal cord than over the volume of a more compact organ such as a kidney. Lastly, the spinal cord is eccentrically placed, being a posterior rather than a midline structure. When an obese patient is treated with parallel-opposed, anterior-posterior fields on a 6oCo machine, the average dose received by the spinal cord may be significantly higher than the prescribed midplane dose.
DISCUSSION Tolerance doses for human organs can only be determined from human data (3, 4, 5, 11, 12, 14, 20). It is therefore necessary that, when complications are reported,
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the data are presented in such a way that these tolerance doses can be determined. Not only should the prescribed dose be stated, but the actual dose distribution received by the damaged organ should be ascertained and presented. Merely stating the prescribed dose could be misleading, as complications are most likely to occur in precisely those cases where deviations from the prescribed dose, within the damaged organ, are the greatest. For reasons previously stated, these considerations are most important for the spinal cord. Thus, when cases of radiation myelitis are reported, the dose distribution received by the spinal cord should be given. This is not always done, even in recent articles. McCunniff and Liang ( 12) reported on two patients who developed symptoms of radiation myelitis in a series of 652 patients treated with radiation for head and neck cancer. One patient who developed myelitis received a dose of 5000 cGy in 30 fractions whereas the second received 6000 cGy in 30 fractions. Although it is not explicitly stated in the paper, it is likely, from the round numbers, that these are the prescribed doses rather than spinal cord doses (12). McCunniff and Liang present several reasons why an area of the first patient’s spinal cord may have received a dose that was substantially higher than the stated 5000 cGy in 30 fractions, but they make no attempt to determine how much higher. Dische et al. (5) have documented the dangers of treating the human spinal cord with large dose fractions. Al-
July 1990, Volume
19, Number
1
though they determine the incidence of myelitis as a function of the spinal cord dose, they do not explicitly state whether this is the cord dose in a particular plane, an average cord dose or the highest dose received by the spinal cord. Cohen et al. (4) have documented the sensitivity of the human spinal cord to neutron therapy. They determined both the central cord dose and the maximum cord dose, although in their scatter plot of cord dose versus treatment time they used the former rather than the latter. Note that in their table of six patients who developed transient paresthesia, where both the central cord dose and the maximum cord dose were listed for each patient, the latter exceeded the former by as much as 55%. In any clinical situation where the spinal cord is in the held, it would be a good idea to keep track of the dose distribution received by the spinal cord and of the maximum daily dose received anywhere along the cord. The spinal cord should be excluded from the field before this maximum dose exceeds cord tolerance. Not only would such a procedure be likely to reduce the incidence of myelitis, but, in those very rare cases where myelitis did occur, the data would be more valuable. In particular, it might be possible to determine the minimum value of D, at which this complication occurs in humans. Then, if the LQ model is correct, one could feel confident that, as long as this value of D, is not exceeded anywhere along the spinal cord, radiation myelitis would be very unlikely to occur.
REFERENCES I. Ang, K.; Thames, H.; van der Kogel, A.; van der Schueren, E. Is the rate of repair of radiation induced sublethal damage in the rat spinal cord dependent on the size of the dose per fraction? Int. J. Radiat. Oncol. Biol. Phys. 13:557-562; 1987. 2. Barendsen, G. W. Dose fractionation, dose rate and isoeffect relationships for normal tissue responses. Int. J. Radiat. Oncol. Biol. Phys. 8: 198 l- 1997; 1982. 3. Boden, G. Radiation myelitis of the cervical spinal cord. Br. J. Radio]. 21:464-469; 1948. 4. Cohen, L.; Ten Haken, R.; Mansell, J.; Yalavarthi, S.; Hendrickson, F.; Awschalom, M. Tolerance of the human spinal cord to high energy P. (66) Be (49) neutrons. Int. J. Radiat. Oncol. Biol. Phys. 11:743-749; 1985. 5. Dische, S.; Warburton, M.; Saunders, M. Radiation myelitis and survival in the radiotherapy of lung cancer. Int. J. Radiat. Oncol. Biol. Phys. 15:75-8 1; 1988. 6. Douglas, B.; Fowler, J. The effect of multiple small doses of X rays on skin reactions in the mouse and a basic interpretation. Radiat. Res. 66:40 l-426; 1976. 7. Fischer, J.; Moulder, J. The steepness of the dose-response curve in radiation therapy. Radiology 117: 179- 184; 1975. of inhomogeneous 8. Goiten, M. Causes and consequences dose distributions in radiation therapy. Int. J. Radiat. Oncol. Biol. Phys. 12:70 l-704; 1986. 9. Hopewell, J.; Morris, A.; Dixon-Brown, A. The influence of field size on the tolerance of the rat spinal cord to single doses of X rays. Brit. J. Radiol. 60:1099-l 108; 1987. 10. Hopewell, J.; van der Kogel, A. Volume effect in spinal cord (author’s reply). Brit. J. Radiol. 61:973-975; 1988.
1 I. Lambert,
12.
13.
14.
15.
16.
17.
18.
P. Radiation myelopathy of the thoracic spinal cord in long-term survivors treated with radical radiotherapy using conventional fractionation. Cancer 4 1: 175 I- 1760; 1978. McCunniff, A.; Liang, M. Radiation tolerance at the cervical spinal cord. Int. J. Radiat. Oncol. Biol. Phys. 16:675-678; 1989. Paliwal, B.; Griem, M., eds. Treatment planning: a categorical course in radiation therapy. Oak Brook, IL: Radiological Society of North American; 1986. Reinhold, H.; Kaalen, J.; Unger-Gils, K. Radiation myelopathy of the thoracic spinal cord. Int. J. Radiat. Oncol. Biol. Phys. 1:65 l-657; 1976. Rubin, P. Late effects of chemotherapy and radiation therapy: A new hypothesis. Int. J. Radiat. Oncol. Biol. Phys. 10:5-34; 1984. Sheline, G.; Wara, W.; Smith, V. Therapeutic irradiation and brain injury. Int. J. Radiat. Oncol. Biol. Phys. 6: 12 151228; 1980. Thames, H.; Withers, J. R.; Peters, L.; Fletcher, G. Changes in early and late radiation response with altered dose fractionation: implications for dose-survival relationships. Int. J. Radiat. Oncol. Biol. Phys. 8:219-226; 1982. van der Kogel, A. Effect of volume and localization on rat spinal cord tolerance. In: Fielden, E., Fowler, J., Hendry, J., Scott, D., eds. Proceedings ofthe 8th International Congress of Radiation Research, London: Taylor & Francis; 1987:352.
Biological effect of inhomogeneous dose distributions 0 R. J. YAES
19. van der Kogel, A. Radiation tolerance of the rat spinal cord: time dose relationship. Radiology 122:505-509; 1977. 20. Wara, W.; Phillips, T.; Sheline, G.; Schwade, J. Radiation tolerance of the spinal cord. Cancer 35: 1558-l 562; 1975. 21. White, A.; Hornsey, S. Radiation damage to the rat spinal cord: the effect of single and fractionated doses of X rays. Brit. J. Radial. 51:515-523; 1978. 22. Withers, H. R.; Taylor, J. Volume effect in spinal cord. Brit. J. Radial. 61:973; 1988. 23. Withers, H. R.; Taylor, J.; Maciejewski, B. Treatment volume and tissue tolerance. Int. J. Radiat. Oncol. Biol. Phys. 14:751-760; 1988. 24. Withers, H. R.; Thames, H.; Peters, L. A new isoeffect curve for change in dose per fraction. Radiother. Oncol. 1: 187191; 1983.
201
25. Wolbarst, A. B.; Chin, L. M.; Svenson, G. K. Optimization of radiation therapy: integral response of a model biological system. Int. J. Radiat. Oncol. Biol. Phys. 8: 1761-1770; 1982. 26. Yaes, R. J. Some implications of the linear-quadratic model for tumor control probability. Int. J. Radiat. Oncol. Biol. Phys. 14:147-157; 1988. 27. Yaes, R. J.; Gorman, S.; Berner, B.; Kalend, A.; Maruyama, Y. Comment on the recent results of Hopewell, Morris and Dixon-Brown on radiation myelitis produced by irradiation of very short segments of rat spinal cord. Brit. J. Radiol. 61:967-969; 1988. 28. Yaes, R. J.; Kalend, A. Local stem cell depletion model for radiation myelitis. Int. J. Radial. Oncol. Biol. Phys. 14: l2471259; 1988.
19. pp.
0360.3016/90 $3.00 + .Xl Copyright C 1990 Pergamon Press plc
203-207
0 Oncology Intelligence THE BIOLOGICAL EFFECT OF INHOMOGENEOUS DOSE DISTRIBUTIONS IN FRACTIONATED RADIOTHERAPY ROBERT J. YAES, Sc.D., M.D. Department
of Radiation Medicine, University of Kentucky Medical Center, Lexington, KY 40536-0084 U.S.A.
The linear quadratic (LQ) model is applied to an organ receiving a fractionated course of radiotherapy with an inhomogeneous dose distribution. It is shown that the gradient in the extrapolated response dose (ERD) will be steeper than the gradient in the physical dose. This effect will be greatest for an organ with a small a/@ ratio treated with large dose fractions. Clinical implications are discussed with an emphasis on radiation myelitis. Radiation myelitis, Linear quadratic model, Late effects, Normal tissue damage.
effect, since both the fraction size and the total dose will be higher. This double effect can be easily quantified by using the linear-quadratic (LQ) model. Since, in the LQ model, the response to changes in fraction size will depend on the a/p ratio (2, 6, 17, 24, 26) this response will vary from organ to organ. For simplicity, the effect of an inhomogeneous dose delivered to a single organ is assessed. In an organ we can choose a point at or near the center of the organ as the origin and take the dose at this point D (a) given in N fractions of size d (6) to be the “average” dose received by the organ. Of course
INTRODUCTION In an animal experiment ( 1,6,9, 18, 19,2 1) whose object is to determine normal tissue tolerance, a uniform dose of radiation is delivered to an entire organ or to a specified volume of tissue. Total doses, fraction sizes and treatment volumes can be varied in a systematic way, using a large number of individual animals, so that detailed dose-response curves can be obtained. Models such as the linearquadratic (LQ) model (2,6) can be tested and the parameters of these models can be accurately determined. It is not clear however, how normal tissue tolerance data, obtained from experimental animals can be extrapolated to man (1). In a clinical situation, our objective is not to deliberately induce normal tissue damage, but, on the contrary, to attempt to cure a patient’s cancer while minimizing the liklihood of clinically significant complications. Although it is intended that a uniform dose be delivered to the entire treatment volume, when large fields must be used to include all areas of potential microscopic tumor extension, inhomogeneities invariably occur. As a result, for each daily treatment, small volumes of normal tissue may receive doses that are consistently higher than the prescribed dose. Goiten (8) has considered the effect of inhomogeneties on the liklihood of tumor control so we will limit our discussion to the problem of normal tissue damage. METHODS
AND
D (5) = N d (-6).
Any other point in the organ can be specified by the vector 51= (x, y, z) and the dose at the point 51is D (si) = Nd(SI)
(2)
where d (2) is the fraction size received at point 51(Fig. 1). It is explicitly assumed that the same isodose distribution is valid for each fraction. If the variation in dose and fraction size within the organ is small, we can write D (2) = D (5) + AD(Z) d (X) = d(Ti) + Ad (X)
MATERIALS
AD(Z) = NAd(St).
Application of the linear-quadratic model to an inhomogeneous dose distribution If a volume of normal tissue receives a daily dose that is higher than the prescribed dose, there will be a double
Accepted for publication
(1)
(3)
We assume that Ad(Z) is small compared to d(a) and thus AD@) is small compared to D(a).
24 January 1990. 203
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1. J. Radiation Oncology 0 Biology 0 Physics
July 1990, Volume 19, Number 1
Since D(Z) = Nd(Z), Eq. (8) becomes D,(Z)
Fig. 1. Non-uniform dose distribution. The origin o within an organ receives a dose D in N equal fractions of size d. Any other point within the organ can be specified by the vector 2 = (x, y, z). The dose at the point TIis D(Z) given in N equal fractions of size d(Z). We explicitly assume that the same isodose distribution is valid for each fraction.
In the linear quadratic model, organ damage is assumed to result from the depletion of a target stem cell population (6, 15) whose survival probability, S has the form S = eeE,
E = cuD + @D2.
(4)
If the total dose D is given in N equal fractions of size d, and if sufficient time is allowed for complete repair of sublethal damage between fractions, instead of Eq. (4) we would have (6, 17) E = aD + /3Dd.
(5)
It is also an assumption of the LQ model that “isoeffect equals isosurvival,” that is, volumes of tissue that have the same value of E and hence the same value of S will have the same level of damage (6). It is easy to show that the equation for the isoeffect curves in the d-D plane (the values of d and D that produce the same value of E and hence of S) has the form (6, 17, 24, 26) D(d + 6) = 6D,
(6)
D,
(7)
or = D(d + 6)/6
Response Dose, where D, = E/a is the Extrapolated (ERD) and 6 = CX/~is the (Y/P ratio and has the dimensions of dose. We wish to write Eq. (7) for the origin Ti and the point x’. For simplicity, we will define D = D(a), D, = D,(G), d = d(a). Then the equation at 6 will be identical to Eq. 7. At x we will have
D,(g)
= D(Sl)(d(ST) + 6)/a.
(8)
= D(Z) + D’(?)/NS.
(9)
D(Z) is the spatial dose distribution. The value of D(Z) at point Z is the physical dose that would be measured by a small ionization chamber placed at the point TI. D,(Z) is a measure of the cell survival probability, S, at the point Z and can therefore be thought of as the “biologically effective” dose distribution. D,(Z) will depend on the fraction size and on the a//3 ratio, as well as on the physical dose distribution D(Z). Curves along which D(Z) remains constant are the isodose curves. From Eq. 9, it is clear that D,(X) will also be constant along such curves. Thus, as one might expect, isodose curves are also isoefict curves. However, the gradient in biologically effective dose will be greater than the gradient in physical dose as we can see by differentiating Eq. 9. + D,(x)
= [(2d t 6)/6] - + D(x).
(10)
If we again assume that the variation in dose over the treatment field is small, then, for the change in dose as we go from the origin 6 to the point 51we would have A D(Z) = 2-G A D,(Z)
= Sr.?
It is then easy to see by dividing using Eqs. 8, 10 and 11,
D(SI)
(11)
D,(Z).
(12)
Eq. 12 by D,(X)
and
A D, (3/D, C-4 = [(2d + 6)/(d + S)] - [A D(s;)/D(s;)].
(13)
The significance of Eq. 13 can best be illustrated with some simple examples. Consider a late reacting tissue with an a/p ratio of 2 Gy treated with 2 Gy fractions. The proportionality factor, (2d + 6)/(d + a), is then 3 = 1.5. Thus, in this case, if the dose D(Z) is 10% higher at ? than at i3, the biologically effective dose at this point would be 15% higher. If the same organ were treated with a hyperfractionated regimen with 1 Gy fractions, a 10% increase in physical dose would lead to a 13% increase in biologically effective dose whereas for a hypofractionated regimen with 5 Gy fractions, the same 10% increase in physical dose would give a 17% increase in biologically effective dose. For an early reacting tissue, with an (Y/P ratio of 10 Gy treated with 2 Gy fractions, a 10% increase in physical dose would result in less than a 12% increase in biologically effective dose. Eq. 13 implies that if there is a volume of normal tissue that consistently receives a slightly higher dose than the prescribed dose, the effect on this tissue will be magnified by a factor of (2d + 6)/(d + 6). This magnification factor
Biological effect of inhomogeneous dose distributions 0 R. J. YAES
will be greatest for an organ with a small CY//~ratio treated with large dose fractions. Thus, there is a second beneficial effect of hyperfractionation in addition to the direct sparing of late reacting tissues ( 17) with small (Y/Pratios. The effect of any “hot spots” involving these same organs will be minimized. The extrapolated response dose, D,(Z), is proportional to the logarhythm of the survival probability S for the target stem cell population (2). The relationship between the survival probability, S and the probability, P, of organ damage, however, is complex and depends on the details of the functional anatomy of the organ in question. A useful concept is the analogy with electrical circuits (25) whereby many organs can be considered to be either “series organs” or “parallel organs.” A “parallel organ” consists of a large number of structurally and functionally independent subunits all performing the same function “in parallel”, so that the output of the organ as a whole is the sum of the outputs of the individual subunits. If the organ is damaged, the degree of damage will be proportional to the number of subunits that have been inactivated. Examples of parallel organs would be the kidney composed of nephrons or the lung composed of alveoli. In a series organ, tissue continuity is essential for organ function. Disturbance of that continuity by destruction of even a very small volume of tissue could cause the organ to cease functioning entirely. An example of a series organ would be the spinal cord (23, 27, 28). A model that would relate the probability of organ damage to the cell survival probability has been discussed in detail elsewhere (10, 22, 23, 27, 28). There is no need to repeat that discussion here. For the purpose of this discussion a simple approximation might suffice. For organs with steep dose-response curves (7) we can assume that for each type of tissue there is an extrapolated tolerance dose, (2) D’, . When the extrapolated response dose D,(Z) exceeds the extrapolated tolerance dose D’, in a given volume of tissue, the tissue within that volume will be rendered non-functional. Hence if we can determine the spatial distribution of the extrapolated response dose within an organ, D,(Z), and if we know the extrapolated tolerance dose D’, for that organ, we should be able to predict how much damage that organ would suffer. Computerized treatment planning systems are now available that can use digitalized CAT scan data to determine the dose distribution in 3-dimensional space within any organ ( 13). If the (Y/Pratio for the organ is known and the fraction size is specified, it would be a simple matter to modify these systems so as to calculate the biologically effective dose distribution as well. RESULTS Radiation myelitis
The organ for which these considerations will be most important will be the spinal cord for several reasons.
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Firstly and probably most important, the spinal cord (1, 3, 4, 5, 9, lo, 11, 12, 14, 16, 18, 19, 20, 21, 22, 27, 28) can be considered all to be a “series organ.” Destruction of a very small volume if tissue will lead to a devastating complication (22,25,27,28). Destruction of a thin “slice” of spinal cord tissue, by radiation or by any other means, can result in cord transection, complete loss of all neurological function below the level of the lesion. Thus, an inhomogeneity or overlap that causes the overdosing of a very small volume of spinal cord tissue can have clinically significant results. Secondly, the a/P ratio for spinal cord of between 1 and 2 Gy is small even for late reacting tissue (1). Thus, for a given fraction size the magnification factor of (2d + 6)/(d + 6) will be larger for the spinal cord than for most other organs. It is for this reason that overlap on the spinal cord between adjacent fields that are treated simultaneously could be more dangerous than a similar overlap between fields that are treated sequentially (retreatment). When there is overlap between fields that are treated simultaneously, both the fraction size and the total dose will be increased in the area of overlap. When there is overlap between fields that are treated sequentially, there will be an increase in the total dose but not in the fraction size in the area of overlap. (There are, however both experimental and theoretical reasons to believe that very small areas of overlap, less than 0.5 cm, could be tolerated (9, 10, 18, 22, 27, 28)). Thirdly, the dose response curve for radiation myelitis is very steep, (7, 19, 2 1) so that even a small change in dose can result in a large change in the complication probability. Thus, when the single dose of radiation received by the rat lumbar spinal cord (19) is increased by 5% from 1900 to 2000 cGy, the probability of myelitis increases from 0% to 100%. Even when the dose to the lumbar cord (19) is given in 10 equal fractions, a 20% increase in dose from 4500 to 5500 cGy can also increase the complication probability from 0% to 100%. Although detailed dose-response curves for the human spinal cord are not available, it is possible that they could be comparably steep. Fourthly, the shape of the spinal cord is long and thin. There is likely to be a greater variation in dose over the length of the spinal cord than over the volume of a more compact organ such as a kidney. Lastly, the spinal cord is eccentrically placed, being a posterior rather than a midline structure. When an obese patient is treated with parallel-opposed, anterior-posterior fields on a 6oCo machine, the average dose received by the spinal cord may be significantly higher than the prescribed midplane dose.
DISCUSSION Tolerance doses for human organs can only be determined from human data (3, 4, 5, 11, 12, 14, 20). It is therefore necessary that, when complications are reported,
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the data are presented in such a way that these tolerance doses can be determined. Not only should the prescribed dose be stated, but the actual dose distribution received by the damaged organ should be ascertained and presented. Merely stating the prescribed dose could be misleading, as complications are most likely to occur in precisely those cases where deviations from the prescribed dose, within the damaged organ, are the greatest. For reasons previously stated, these considerations are most important for the spinal cord. Thus, when cases of radiation myelitis are reported, the dose distribution received by the spinal cord should be given. This is not always done, even in recent articles. McCunniff and Liang ( 12) reported on two patients who developed symptoms of radiation myelitis in a series of 652 patients treated with radiation for head and neck cancer. One patient who developed myelitis received a dose of 5000 cGy in 30 fractions whereas the second received 6000 cGy in 30 fractions. Although it is not explicitly stated in the paper, it is likely, from the round numbers, that these are the prescribed doses rather than spinal cord doses (12). McCunniff and Liang present several reasons why an area of the first patient’s spinal cord may have received a dose that was substantially higher than the stated 5000 cGy in 30 fractions, but they make no attempt to determine how much higher. Dische et al. (5) have documented the dangers of treating the human spinal cord with large dose fractions. Al-
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though they determine the incidence of myelitis as a function of the spinal cord dose, they do not explicitly state whether this is the cord dose in a particular plane, an average cord dose or the highest dose received by the spinal cord. Cohen et al. (4) have documented the sensitivity of the human spinal cord to neutron therapy. They determined both the central cord dose and the maximum cord dose, although in their scatter plot of cord dose versus treatment time they used the former rather than the latter. Note that in their table of six patients who developed transient paresthesia, where both the central cord dose and the maximum cord dose were listed for each patient, the latter exceeded the former by as much as 55%. In any clinical situation where the spinal cord is in the held, it would be a good idea to keep track of the dose distribution received by the spinal cord and of the maximum daily dose received anywhere along the cord. The spinal cord should be excluded from the field before this maximum dose exceeds cord tolerance. Not only would such a procedure be likely to reduce the incidence of myelitis, but, in those very rare cases where myelitis did occur, the data would be more valuable. In particular, it might be possible to determine the minimum value of D, at which this complication occurs in humans. Then, if the LQ model is correct, one could feel confident that, as long as this value of D, is not exceeded anywhere along the spinal cord, radiation myelitis would be very unlikely to occur.
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