2.1.

Link to Single-Pulse Dose–Response Curve

Solving Eq. (2) for $p$,

Equations (2–4) hold, in general, for exposures consisting of identical pulses, regardless of the specific dose–response curves used for $p$ and $P$. A particularly interesting feature of Eq. (4) is that it suggests that the SP dose–response curve could be determined by measurement of the MP ${\mathrm{ED}}_{50}$. For example, $n=4$ gives $p=0.16$, so the ${\mathrm{ED}}_{50}$ for a four-pulse exposure corresponds to the ${\mathrm{ED}}_{16}$ for an SP exposure. Assuming a log-normal response, the SP probit slope can be calculated from these values since,⁴

2.2.

Application to Spot Size Dependence

Menendez et al.¹ mentioned that the PSM would also apply to exposures separated in space as well as time. The authors were referring to separate laser exposures to different areas of the retina, but the model can also be applied to the spatial distribution of a single exposure. An irradiated area can be considered as a collection of smaller subareas. If each subarea within the irradiated area is statistically independent from the others, then the probability of damage occurring at each subarea will sum to give the net probability of damage occurring. This requirement of statistical independence certainly does not exist for long exposures in which heat can conduct through the irradiated volume. But, if the exposure time is short enough, such that heat does not have time to conduct through the volume during the exposure, then these points can be considered statistically independent from each other during the time of the exposure. It is generally accepted that microcavitation causes damage for exposure durations $<10\mu \mathrm{s}$ and that heat does not significantly conduct through the irradiated volume during this time.

Therefore, if a laser exposure covers an area $A$ that contains $m$ possible microcavitation sites, the probability of the exposure causing damage should just be the probability that one of the sites produces microcavitation. Assuming that the density, $\sigma$, of microcavitation “seeds” (sites which are capable of producing microcavitation) is uniform, the number of seeds exposed by a uniform laser beam with area $A$ will be $m=\sigma A$. The probability of damage occurring within the irradiated area will be

2.3.

Effect on Probit Slope

A closer look at Eq. (3) indicates that the dose–response curve will become sharper with more pulses. For log-normal distributions, the probit slope is given by Eq. (5). If we consider the case of $n=4$, then the SP ${\mathrm{ED}}_{16}$ gives the MP ${\mathrm{ED}}_{50}$, and the SP ${\mathrm{ED}}_{37}$ gives the MP ${\mathrm{ED}}_{84}$. Therefore,

3.1.

Probit Slope for Small-Spot In-Vivo Data

Note: In this section, we will use “actual” and “measured” to refer to the true physical value of a statistical quantity and the value obtained for the quantity by measurements, respectively. The measured value is an approximation of the actual value.

We have shown in Sec. 2.1, that the slope of the SP dose–response curve is directly related to the MP ${\mathrm{ED}}_{50}$ trends. For example, based on Eqs. (5) and (10), the ${\mathrm{ED}}_{50}$ for a four-pulse exposure could be written in terms of the SP slope directly:

Fig. 2

The MP damage threshold reduction predicted by the PSM applied to log-normal dose–response curves with various slopes. The reduction is greatest when the slope is small and diminishes for larger slopes. Dashed line indicates the $n^{-1/4}$ trend.

Sliney and Lund^3,7 have argued that the observed $n^{-1/4}$ trend may simply be an artifact of experimental method. They have stated that the PSM supports this based on the relationship between the SP probit slope and the MP reduction. However, this specific claim (involving the PSM) is invalid, and we wish to correct it here.

They argue that it is very difficult to detect small retinal lesions in the eye and that because of this, the SP dose–response curve for small-spot exposures is affected by experimental uncertainties more so than large-spot exposures. Experimental uncertainties will cause a measured dose–response curve to have a lower probit slope (and higher ${\mathrm{ED}}_{50}$) than the actual dose–response curve.⁸ They point to the large-spot in-vivo and small-spot explant data in which both have larger probit slopes and smaller MP effects, as evidence that the small-spot in-vivo MP reduction would be much smaller if the actual probit slope could be measured.^3,7 However, if a measured probit slope is smaller than the actual probit slope, it does not follow that the observed MP reduction would be larger.

The PSM determines the actual MP dose–response curve from the actual SP dose–response curve because it is the actual probabilities that are being summed. Equation (2) is a relationship between the underlying probability distributions. Experimental uncertainties limit the ability to observe these underlying distributions, but they do not change this relationship. Therefore, one cannot suggest that if the actual SP probit slope was determined, $\beta$ would be larger and the observed MP reduction would be smaller. If anything, the PSM would suggest that the actual probit slope is small based on the observed MP reduction.

For example, imagine that two different methods are used to observe the same experiment in which the SP damage threshold and the 1000-pulse damage threshold are measured. One includes significant experimental uncertainties, the other does not. The two methods will not agree on the ${\mathrm{ED}}_{50}$ values for each exposure, the inaccurate method will measure a higher ${\mathrm{ED}}_{50}$ for both the SP and 1000-pulse exposures. However, the actual ${\mathrm{ED}}_{50}$ does not depend on the measurement method and is the same for both methods. Now assume that the accurate method measures no difference between the SP and 1000-pulse ${\mathrm{ED}}_{50}$. We would not expect the inaccurate method to measure a significant decrease in the 1000-pulse ${\mathrm{ED}}_{50}$ simply because the probit slope obtained by it for the SP exposure is small.

At this point, we would like to explicitly state that we are not disputing the conclusions of Lund and Sliney,^3,7 specifically that the observed MP reduction in small-spot in-vivo data is an effect caused by the difficulty in observing small-spot lesions. We only wish to clarify that the PSM does not play a role in this. The problem is that this would not be a probit-slope effect but an ${\mathrm{ED}}_{50}$ effect. In the example given above, Lund and Sliney² would argue that the inaccurate method will not measure a significant decrease in the 1000-pulse ${\mathrm{ED}}_{50}$, but instead will measure a significant increase in the SP ${\mathrm{ED}}_{50}$. Therefore, it will appear as if a significant reduction in threshold has occurred from an SP exposure to a 1000-pulse exposure, even if none exists.

If this is in fact the case, then the MP effect is caused by the inability to detect small lesions causing measured ${\mathrm{ED}}_{50}$ values to be larger than the actual values. The continued reduction in ${\mathrm{ED}}_{50}$ with more pulses must be due to an increased ability to detect lesions (so that the measured ${\mathrm{ED}}_{50}$ is closer to the actual, lower ${\mathrm{ED}}_{50}$) and the apparent connection between the small-spot in-vivo probit slope and $n^{-1/4}$ is purely coincidental.

Finally, we note that the PSM predicts that the actual probit slope for MP dose–response curves should increase with the number of pulses, but it does not require that the measured slope do so. Again, experimental uncertainties may severely limit the ability to measure an accurate slope, but the PSM will still predict a decrease in ${\mathrm{ED}}_{50}$ based on the actual SP probit slope. If, on the other hand, the experimental uncertainties are directly responsible for the MP reduction, then the measured probit slope would necessarily depend on the number of pulses. Any observed decrease in ${\mathrm{ED}}_{50}$ due to an increased lesion visibility should be accompanied by an increase in probit slope. It would be interesting to examine the MP dependence of the measured probit slope to see if a strong correlation has been observed.

3.2.

Single-Pulse Threshold Dependence on Spot Size

The dependence of the SP ${\mathrm{ED}}_{50}$ on spot size should be similar to that of MPs, but direct application of the PSM is more difficult. Equation (6) requires both the dose–response curve for a single microcavitation seed ($\sigma A=1$) and the microcavitation seed density ($\sigma$) to be known.

However, just as a 100-pulse exposure can be treated as ten 10-pulse exposures, an irradiated area can be treated as multiple smaller irradiated areas. And just as the PSM can be applied to the 10-pulse dose–response curve to produce the 100-pulse dose–response curve, we can apply the PSM to the dose–response for any irradiated area to produce the dose–response curve for larger irradiated areas.

Let $A_0$ be some irradiated area for which the dose–response curve, $P_0(d)$, has been measured. From Eq. (6), we get:

Figure 3 shows the threshold dependence on irradiated diameter, based on the PSM, for uniform beam profiles (flat top). However, an interesting comparison to experiment can be made if we follow the work of Menendez¹ and Lund^5,7 to derive a “correction factor” for the irradiated diameter analogous to the MP $n^{-1/4}$ correction factor. Starting with Eq. (6) and applying the approximations used by Menendez and Lund leads to the following relationship:

Fig. 3

The threshold dependence on irradiated diameter, predicted by the PSM applied to log-normal dose–response curves with various slopes. Note that the $x$ axis is the relative diameter, for a $25\mu \mathrm{m}$ baseline, the right end of the $x$ axis would correspond to $2500\mu \mathrm{m}$. Dashed lines indicate the correction factor approximation from Eq. (15) for each slope.

Lund et al.¹⁰ re-examined the relationship between the damage threshold retinal radiant exposure ($\mathrm{J}{\mathrm{cm}}^{-2}$) and retinal irradiated area, analyzing all available data in the literature, and noted that thermal models could not account for the threshold dependence on irradiated area for short pulses ($<20\mu \mathrm{s}$). For thermal damage, the threshold retinal radiant exposure is expected to increase for smaller beams because heat can conduct out of the irradiated volume. However, if the energy is delivered in a shorter time than it takes heat to conduct out (the thermal confinement time), there should be no increase in threshold for small beams. The experimental data examined by Lund indicate that the retinal radiant exposure threshold does increase for smaller irradiated areas, even for very short exposure times.

This is puzzling, even though we now expect the damage to be caused by microcavitation, because for short-pulse exposures, there should not be enough time for the exposure at the edge of an irradiated area to influence the threshold at the center of the irradiated area. The threshold for causing microcavitation at the center of the area should not depend on the size of the area. The PSM may provide an explanation for this, which is that the probability of damage occurring at the center of the area does not increase, but that the overall probability of damage occurring somewhere in the beam does.

In their re-examination, Lund et al.¹⁰ showed that if the damage thresholds, expressed as a radiant exposure, were plotted versus irradiated diameter on a log-log plot, then a linear line could be fit to most datasets. In other words, each dataset could be approximated by the equation $H_r=k{D}^S$, where $H_r$ is the threshold radiant exposure at the retina and $D$ is the retinal irradiance diameter. $S$ is the slope of the best fit line, on log-log scale, and the value of $S$ that best fit each set depended on the exposure time. Longer exposure times tended to have a large negative value (indicating a large decrease in threshold with increasing diameter) while shorter exposure times tended to have a smaller negative value (indicating a small decrease in threshold with increasing diameter).

For pulses longer than $20\mu \mathrm{s}$, the value of $S$ for any given exposure time was approximated well by the function $S(t)=(-0.233\log (t)+1)$. However, for pulses shorter than $20\mu \mathrm{s}$, no such trend appeared. In addition, the dependence on retinal irradiated diameter for exposures longer than $20\mu \mathrm{s}$ was consistent with thermal model predictions, while the dependence for shorter exposures was not. The thermal model predicted no dependence on the irradiated diameter ($S=0$), but the data showed a decrease in threshold with diameter ($S<0$).

The observed threshold trends for exposure durations below $20\mu \mathrm{s}$ do not match the predictions made by thermal models because microcavitation is responsible for damage for these short times, and the PSM would apply. Rewriting the approximate function for $H_r$ in terms of the retinal irradiance area $A$ gives $H_r\propto A^{S/2}$. Comparing this to Eq. (15) implies that $-1/\beta =S/2$. The values for $S$ observed by Lund et al.¹⁰ for exposures below $20\mu \mathrm{s}$ ranged between 0 and $-1$. This would require the probit slope for $P_0$ in Eq. (14) to be 3 or greater. This is not a very narrow range but is certainly in the range of observed values. For example, a slope of 11 would give a value of $S=-0.2$. More important than the specific value of $S$ is the fact that the damage threshold radiant exposure still depends on the irradiated diameter, even for short pulses where the thermal model predicts no dependence. Now, it should be noted that there are other possible explanations for this (for example, the minimum beam diameter may actually be much larger than is assumed), but this is consistent with the PSM.

Lund et al.¹⁰ also noted that the thermal model predictions agreed with the shorter pulse data better when the irradiated area was large but was increasingly less predictive for irradiance diameters $<100\mu \mathrm{m}$. This too is consistent with our understanding of microcavitation damage. It is still possible to cause thermal damage during short-pulse durations. It is just that the threshold for causing microcavitation is lower, and the observed damage threshold will be the lesser of the two. Therefore, if the thermal damage threshold (as a radiant exposure) is somehow lowered, it can be observed. This can be achieved by increasing the thermal relaxation time, the time it takes the irradiated tissue to cool back down after an exposure. Increasing the irradiated diameter will increase the thermal relaxation time, which can bring the thermal damage threshold below the microcavitation threshold. If the thermal relaxation time is increased, the thermal damage threshold will be observed at lower exposure durations.

3.3.

Multiple-Pulse Thresholds for Large Spot Sizes

In the PSM, the reduction in ${\mathrm{ED}}_{50}$ for an MP exposure is directly related to the slope of the SP dose–response curve. If the (actual) slope of the SP dose–response curve increases, the MP reduction will be smaller.

The observed ${\mathrm{ED}}_{50}$ trends for large spot MP exposures exhibit this behavior. The MP effect on damage thresholds seems to be reduced for large spot size, and in some observations, there appears to be no reduction in the damage threshold.² The PSM provides an explanation for this. As the irradiated area increases, the SP dose–response curve will become more sharp (see Sec. 2.3), and the MP effect will decrease. Note that this prediction is counter to that of thermal damage models, which predict that the MP effect should be stronger for larger spot sizes.¹¹ This may provide an additional test (along with the independence of MP thresholds on duty cycle) to differentiate microcavitation and thermal damage mechanisms. The observed $n^{-1/4}$ trend was based on small-spot exposures (collimated beam into the eye), which are more likely to show an MP effect.