First measurements of radon-220 diffusion in mice tumors, towards treatment
planning in diffusing alpha-emitters radiation therapy
Guy Heger1,∗, Mirta Dumančić1,∗,†, Ishai Luz2, Maayan Vatarescu2, Noam
Weizman1,3, Brian W. Miller4, Tomer Cooks2,‡, Lior Arazi1,‡
1Unit of Nuclear Engineering, Faculty of Engineering Sciences, Ben-Gurion
University of the Negev, P.O.B. 653 Be’er-Sheva 8410501, Israel
2 The Shraga Segal Department of Microbiology, Immunology, and Genetics, Ben-
Gurion University of the Negev, Be’er Sheva, Israel
3 Oncology Department, Radiation Therapy Unit, Hadassah – Hebrew University
Medical Center, Israel
4 College of Medicine, Department of Radiation Oncology, Department of Medical
Imaging, The University of Arizona, Tucson, AZ 85724 USA
∗ These authors contributed equally
† Current address: Gerald Bronfman Department of Oncology, Faculty of Medicine
and Health Sciences, McGill University, Montreal, Quebec, Canada

‡ Authors to whom correspondence should be addressed. Lior Arazi:
<EMAIL_ADDRESS>Tomer Cooks<EMAIL_ADDRESS>

###### Abstract

Background: Diffusing alpha-emitters radiation therapy (“Alpha-DaRT”) is a new
method for treating solid tumors with alpha particles, relying on the release
of the short-lived alpha-emitting daughter atoms of radium-224 from
interstitial sources inserted into the tumor. Alpha-DaRT tumor dosimetry is
governed by the spread of radium’s progeny around the source, as described by
an approximate framework called the “diffusion-leakage model”. The most
important model parameters are the diffusion lengths of radon-220 and
lead-212, and their estimation is therefore essential for treatment planning.
Purpose: Previous works have provided initial estimates for the dominant
diffusion length, by measuring the activity spread inside mice-borne tumors
several days after the insertion of an Alpha-DaRT source. The measurements,
taken when lead-212 was in secular equilibrium with radium-224, were
interpreted as representing the lead-212 diffusion length. The aim of this
work is to provide first experimental estimates for the diffusion length of
radon-220, using a new methodology.
Methods: The diffusion length of radon-220 was estimated from autoradiography
measurements of histological sections taken from 24 mice-borne subcutaneous
tumors of five different types. Unlike previous studies, the source dwell time
inside the tumor was limited to 30 minutes, to prevent the buildup of
lead-212. To investigate the contribution of potential non-diffusive
processes, experiments were done in two sets: fourteen in-vivo tumors, where
during the treatment the tumors were still carried by the mice with active
blood supply, and ten ex-vivo tumors, where the tumors were excised before
source insertion and kept in a medium at $37^{\circ}$C with the source inside.
Results: The measured diffusion lengths of radon-220, extracted by fitting the
recorded activity pattern up to 1.5 mm from the source, lie in the range
$0.25-0.6$ mm, with no significant difference between the average values
measured in in-vivo and ex-vivo tumors: $L_{Rn}^{in-vivo}=0.40\pm 0.08$ mm vs.
$L_{Rn}^{ex-vivo}=0.39\pm 0.07$ mm. However, in-vivo tumors display an
enhanced spread of activity 2-3 mm away from the source. This effect is not
explained by the current model and is much less pronounced in ex-vivo tumors.
Conclusions: The average measured radon-220 diffusion lengths in both in-vivo
and ex-vivo tumors are consistent with published data on the diffusion length
of radon in water and lie close to the upper limit of the previously estimated
range of $0.2-0.4$ mm. The observation that close to the source there is no
apparent difference between in-vivo and ex-vivo tumors, and the good agreement
with the theoretical model in this region suggest that the spread of radon-220
is predominantly diffusive in this region. The departure from the model
prediction in in-vivo tumors at large radial distances may hint at potential
vascular contribution, which will be the subject of future works.

Keywords: DaRT, Targeted Alpha Therapy, alpha dose calculations,
brachytherapy.
Running title: Radon-220 diffusion measurement in mice tumors.

##  1 Introduction

The radiobiological benefits of alpha particles in cancer treatment are well
established 1, 2, 3. With a linear energy transfer (LET) of $60-230$
keV/$\upmu$m 4, they form dense tracks of ionization, which lead by direct
interaction to irreparable complex DNA damage 5, 6. At the peak ionization
density of their tracks, the cell-killing efficacy of alpha particles is
essentially unaffected by cellular oxygen levels, making them nearly as potent
against hypoxic cells as against oxic ones 7, 8, 9. Furthermore, the range of
alpha particles in water, $\sim 40-90\;\upmu$m, can in principle guarantee the
sparing of healthy tissue. Given the promising therapeutic potential of alpha
particles, multiple preclinical and clinical studies are underway towards
their utilization in cancer treatment, as discussed in several recent
reviews10, 11, 12. Of these, 223RaCl2 treatments for castration-resistant
prostate cancer with bone metastases are already commercial and in widespread
clinical use 13.

Diffusing alpha-emitters radiation therapy (“Alpha DaRT”)14, enables the
treatment of solid tumors using alpha particles. In Alpha DaRT, tumors are
treated with sources carrying low activities of 224Ra below their surface.
Once inside the tumor, the sources are designed to continuously release from
their surface the short-lived daughter atoms of 224Ra: 220Rn, 216Po, 212Pb,
212Bi, 212Po and 208Tl. These spread by diffusion, with possible contribution
from convective effects, creating—primarily through their alpha decays—a
lethal high-dose region measuring several mm in diameter around each source.
The full decay chain, along with the isotopes’ half-lives, decay modes, and
mean alpha particle energies is shown in Figure 1.

Figure 1: The 224Ra decay chain 15.

Alpha DaRT was, and still is, extensively investigated in in vitro and in vivo
preclinical studies on a large number of cancer types, as a stand-alone
treatment 14, 16, 17, 18, 19, in combination with chemotherapy 20, 21, 22, 23,
and as a stimulant of a local and systemic anti-tumor immune response 24, 25,
26, 27, 28, 29, 30, 31. Since 2017, Alpha DaRT has been under clinical
investigation, starting with locally advanced and recurrent squamous cell
carcinoma (SCC) of the skin and head and neck 32. The first-in-human trial,
employed 2 $\mu$Ci 224Ra sources, inserted with a nominal spacing of 5 mm. The
clinical outcomes of this trial showed promising efficacy and safety 33: all
treated tumors (28/28) displayed positive response (shrank by $\sim
30-100\%$), with 78.6% (22/28) exhibiting complete response. Adverse effects
were mild to moderate, with no observable local or systemic radiation-induced
damage to healthy tissue. The alpha dose to all organs, resulting from 212Pb
leaving the tumor through the blood, was calculated to be on the cGy level,
with blood and urine activity measurements largely consistent with the
prediction of a biokinetic model 34. In one patient, untreated lesions shrank
and disappeared when another was treated with Alpha DaRT (in the absence of
any other treatment), suggesting an abscopal effect 35. In a more recent study
on recurrent or unresectable skin cancer, employing 3 $\mu$Ci sources at a
nominal spacing of 4 mm, the complete response rate was 100% (10/10) with only
grade 1 and 2 adverse effects 36.

An in-depth description of the underlying physics of Alpha DaRT37 introduced,
as a first step towards treatment planning, a simplified theoretical approach:
the “Diffusion-Leakage (DL) model”. It was shown that of the six radionuclides
released from the source, only the migration of 220Rn, 212Pb, and—to a lesser
extent—212Bi needs to be modeled, as their respective short-lived daughters
are in local secular equilibrium. It was further shown that the spatial spread
of the alpha dose around an Alpha-DaRT source is dictated by the diffusion
lengths of 220Rn and 212Pb—$L_{Rn}$ and $L_{Pb}$—which are the most critical
parameters for Alpha-DaRT treatment planning. The theoretical and numerical
work concerning the DL model was subsequently extended to realistic source
geometries38 and source lattices39, including a sensitivity analysis that
considered changes in all model parameters, with an emphasis on the diffusion
lengths. These calculations were later corroborated by independent works
employing different finite-element methods 40, 41. Alpha-dose calculations
were recently augmented by a Monte-Carlo-based evaluation of the low-LET
contribution in Alpha DaRT 42.

Measurements of the spatial spread of 224Ra daughters in Alpha-DaRT-treated
mice tumors14, 43, 17, 21, 19, 23, 44 have so far been done $\sim 4-5$ days
after source insertion when 212Pb is in secular equilibrium with 224Ra, and
the spatial activity profile has attained its asymptotic form37. Data
analysis, done on a tumor-by-tumor basis by fitting autoradiography-based
radial or cumulative 212Pb activity profiles 43, or globally by fitting the
effective diameter corresponding to specified dose levels as a function of
source activity37, indicated—for SCC tumors—a mean value of $\sim 0.6$ mm for
the diffusion length, with individual measurements in the range $0.4-1.0$
mm43. This was interpreted as representing the diffusion length of 212Pb,
while the diffusion length of 220Rn—which has never been measured directly—was
estimated to lie in the range $\sim 0.2-0.4$ mm, where the upper limit
corresponds to its expected value in water37, 45. A limited number of
measurements in pancreatic21, melanoma19, prostate19, colon19, 23, and lung17
tumors in mice, indicated a spatial spread which was generally smaller than in
SCC, with unpublished analysis indicating diffusion lengths in the range $\sim
0.3-0.5$ mm. Recent autoradiography measurements of glioblastoma multiforme
tumors in mice showed large variations in spread44, which—when analyzed in
terms of diffusion lengths—corresponded to values ranging $\sim 0.3-1$ mm.

The aim of this work is to provide, for the first time, experimental estimates
of the diffusion length of 220Rn in mice tumors. We show that by limiting the
source dwelling time inside the tumor to $\sim$30 minutes one can extract
$L_{Rn}$ from the recorded activity profile. The study further aims to probe
whether additional effects—in particular blood flow within the
tumor—contribute to the recorded activity spread. This was done by implanting
Alpha-DaRT sources in mice carcinoma before tumor excision (in vivo) and after
(ex vivo). We find higher-than-expected values for $L_{Rn}$, as well as
indications for non-diffusive effects that contribute to the alpha dose at
large distances from the source in vivo but not ex vivo. We discuss the
possible implications of these findings on the usability and limitations of
the DL model for Alpha DaRT treatment planning.

##  2 Methods

###  2.1 General procedure description

Values of $L_{Rn}$ were extracted from autoradiography-measured sections of 24
mice-borne tumors. The data is divided into two sets: in the first, which
contained 14 tumors and labeled in vivo, a single Alpha-DaRT source was
inserted into each tumor, still in the live animal, for 30 minutes. After 30
minutes (the reasoning for selecting this dwelling time is explained in
section 2.5) the tumors were excised and the sources removed. The tumors were
then set to freeze for 1 hour at -80∘C, and subsequently subjected to
10-$\upmu$m-thick histological sectioning (approximately at $90^{\circ}$ with
respect to the source axis) by a LEICA CM 1520 cryostat. Finally, the sections
were fixed with 4% paraformaldehyde for 10 minutes and rinsed twice with PBS
for 10 minutes each time. In the second set of experiments, which contained 10
tumors and labeled ex vivo, the tumors were excised before source insertion.
These tumors (each with a single source inside) were then submerged in PBS and
placed in an incubator for 30 minutes (37∘C, 5% CO2). The rest of the
procedure was the same as for the in vivo experiments (source removal,
followed by freezing, sectioning and rinsing). The alpha particles emitted
from the fixed tumor sections were imaged by the iQID camera 46 (see
description below) and values of $L_{Rn}$ were then extracted from the
resulting images.

###  2.2 Source preparation

Alpha-DaRT sources were produced by Alpha Tau Medical, following the same
manufacturing methods used for clinical trials. The sources are made of
stainless steel 316L cylindrical tubes, with an outer diameter of 0.7 mm,
inner diameter 0.4 mm and length 6.5 mm. Each source was loaded with $\sim$ 3
$\upmu$Ci 224Ra, firmly fixed to the source surface, with a 220Rn desorption
probability37 of $\sim 45\%$.

###  2.3 Cell cultures

The study was conducted using SQ2 (SCC), 4T1 (triple-negative breast cancer),
PANC2 (pancreatic duct adenocarcinoma), MC38 (colon carcinoma) and C4-2
(prostate carcinoma) cell lines. Due to a possible dependence of the diffusion
lengths on the tumor microenvironment and tissue perfusion 47 that could
further change with the tumor mass, the tumors were grown to various masses in
the range of 0.13-1.6 g. The study was approved by the Ben-Gurion University
Institutional Animal Care and Use Committee, and was conducted according to
the Israeli Animal Welfare Act following the guidelines of the Guide for Care
and Use of Laboratory Animals (National Research Council, 1996) [permits no.
IL-78-12-2018(E) and IL-44-08-2021 (E)]. BALB/c, NOD scid gamma (NSG), and
C57BL/6 mice were inoculated subcutaneously into the low lateral side of the
back.

###  2.4 Alpha-particle autoradiography

For each tumor, several central histological sections were cut, usually
$150-300$ $\upmu$m apart. The alpha particles emitted from the tumor sections
were imaged by the iQID camera46 for $15-60$ hours. A full description of the
iQID system is given in Appendix A. Background values were calculated for each
experiment individually by selecting a $4\times 4$ mm2 region clear of tumor
sections, and averaging the number of alpha-particle hits per pixel in that
region. The resulting values were subtracted from all section measurements of
the corresponding experiment. The estimated source location was chosen, in
most cases, according to the center of gravity (COG) of pixel activity in the
section. In some sections where a visible dip was observed within the recorded
high-activity region, the location was selected manually. An example of such a
case is given in Figure 4a, where the COG-estimated location is marked with a
black dashed circle, while the manually chosen one is the blue circle. Next,
the average number of alpha particle counts per pixel, $\langle
N_{\alpha}(r)\rangle$, was calculated in concentric rings around the estimated
source location at increasing radial distances. The width of the rings, $dr$,
was set to 25.14 $\upmu$m, equal to the iQID’s pixel size. The resulting
“number density profile” was then used to estimate $L_{Rn}$ according to the
procedure described in section 2.5.

###  2.5 Estimating the radon-220 diffusion length from measured number
density profiles

Due to the very short half-live of 220Rn ($\sim 1$ min) compared to the iQID
imaging schedule after source removal (many hours), the recorded alpha decays
from tumor sections are in fact those of 212Bi and 212Po, generated in the
decay chain
220Rn$\rightarrow^{216}$Po$\rightarrow^{212}$Pb$\rightarrow^{212}$Bi$\rightarrow^{212}$Po.
Since the measured tumors are frozen at -80∘C immediately after source removal
and fixed with paraformaldehyde immediately after sectioning, redistribution
of 212Pb and 212Bi relative to the decay locations of 220Rn is considered to
be negligible. The reasoning for this relies on the known binding of 212Pb to
large proteins37 (promoting local fixation) and on experimental data
indicating negligible redistribution of 212Bi relative to 220Rn and 212Pb even
in live tumors at body temperature37. Therefore, one can write:

$N_{\alpha}(\textbf{r})\propto\Gamma_{Bi}(\textbf{r})\propto\Gamma_{Rn}(\textbf{r}),$
(1)

where $N_{\alpha}(\textbf{r})$ is the locally-recorded number of alpha
particles per pixel, $\Gamma_{Bi}(\textbf{r})$ is the local specific activity
of 212Bi when the iQID measurement begins, and $\Gamma_{Rn}(\textbf{r})$ is
the local specific activity of 220Rn at the time of source removal from the
tumor. Note that for the relevant specific activities ($<1$ Bq per pixel) the
iQID’s efficiency does not depend on the count rate 46, ensuring linearity
between $N_{\alpha}(\textbf{r})$ and $\Gamma_{Rn}(\textbf{r})$.

#### The theoretical expression of $\Gamma_{Bi}(r)$:

The expression for $\Gamma_{Bi}(r)$ is obtained by solving the DL model
equations. (We write “$r$” rather than “r”) because the calculation assumes
that the activity is recorded in a plane perpendicular to the source axis and
that there is axial symmetry.) While it would seem logical that for this, the
DL equations should be solved for a realistic source geometry (a finite
cylinder), in reality, it appears that choosing a finite line geometry better
represents the actual measured data, as detailed further in the section 3. The
expression of $\Gamma_{Bi}(r)$ for a finite line source is given below, with
the reasoning behind this choice given at the end of the section:

$\Gamma_{Bi}(r)=A\int_{-l/2}^{l/2}\frac{e^{-\sqrt{r^{2}+z^{2}}/L_{Rn}}}{\sqrt{r^{2}+z^{2}}}dz$
(2)

The integral is a sum of terms of the form $e^{-r^{\prime}/L_{Rn}}/r^{\prime}$
(the solution of the DL equations for a point source geometry 37) with
$r^{\prime}=\sqrt{r^{2}+z^{2}}$, each representing a contribution of an
infinitesimal segment $dz$ along the source length $l$, with $z=0$ at the
source mid-plane, and $r$ the radial distance from the source in cylindrical
coordinates.

Figure 2: Complete time-dependent solution (specific activities) of the DL
model equations for a 10-mm-long Alpha-DaRT line source, assuming $L_{Rn}=0.3$
mm and $L_{Pb}=0.6$ mm. The other model parameters are $P_{leak}(Pb)=0.6$,
$L_{Bi}=0.1L_{Pb}$, $\alpha_{Bi}=0$. The specific activities are shown in the
mid-plane of the source, which carries 3 $\upmu$Ci 224Ra, with
$P_{des}(Rn)=0.45$ and $P_{des}^{eff}(Pb)=0.55$. (a) The specific activities
of 220Rn, 212Pb and 212Bi 1.5 mm away from the source over 14 days; (b) Same
as (a), focusing on the first 30 minutes; (c) Specific activity profiles of
220Rn over the first 60 minutes of source dwell time, with the blue curve
corresponding to 30 minutes. The curves for 15 and 60 minutes overlap the blue
one; (d) Same for 212Bi up to 4 days of source dwell time. The solid red curve
denotes the profile at 30 minutes, and the dashed one at 4 days.

Each measured number profile $\langle N_{\alpha}(r)\rangle$ was fitted by the
theoretical expression of $\Gamma_{Bi}(r)$ using least-squares optimization
with $A$ and $L_{Rn}$ as free parameters. The fitted diffusion lengths
$L_{Rn}$ (one for each section) were then averaged over the various sections
of a given tumor for the final result.

#### Short vs. long source dwell times:

It is important to note that Equation (2), with the slope governed by
$L_{Rn}$, holds only for short source dwell times (the time between source
insertion and removal) of up to a few hours. This is illustrated in Figure 2,
which shows the time-dependent solutions of the DL model equations for the
case $L_{Rn}=0.3$ mm, $L_{Pb}=0.6$ mm and $L_{Bi}=0.06$ mm. Panels (a) and (b)
show the time-dependent specific activities of 220Rn, 212Pb, and 212Bi at
$r=1.5$ mm as a function of time, considering long (days, panel a) and short
(minutes, panel b) time scales. While the spatial buildup of 220Rn occurs over
a few minutes, that of 212Pb and 212Bi takes days. In particular, as shown in
(b), the specific activity of 212Bi is two orders of magnitude smaller than
that of 220Rn 30 minutes after source insertion. Panel (c) shows a sequence of
radial specific activity profiles of 220Rn, indicating that $\sim 15$ min
after source insertion, they stabilize on an asymptotic form over the entire
relevant radial range. Panel (d) shows the same for 212Bi: at short time
scales, the radial profile is exponential (linear on a logarithmic scale),
with the same slope as that of 220Rn; over time the profile gradually changes,
with a region of a milder slope (corresponding, here, to the larger diffusion
length of 212Pb) expanding radially outward. Four days after source insertion,
the radial profile of the specific activity of 212Bi becomes linear (on a log
scale) and is completely dominated by $L_{Pb}$. As discussed further in
Appendix B, the slope of the specific activity profile of 212Bi shows a two-
step behavior as a function of time: a short-term step, with a plateau over
$\sim 10-100$ min after source insertion, corresponding to $L_{Rn}$, and a
long-term step, reaching a plateau $\sim 2000$ min ($\sim 1.5$ d) later,
corresponding to $L_{Pb}$. Therefore, autoradiography measurements performed
$\sim 4-5$ d after source insertion provide the dominant diffusion length
$L_{dominant}=\textrm{max}(L_{Rn},L_{Pb})$, without being able to tell if it
corresponds to that of 220Rn or 212Pb. On the other hand, the slope of the
activity profile recorded $\sim 30$ min after source insertion, necessarily
corresponds to $L_{Rn}$. In the case $L_{Rn}>L_{Pb}$ the slope corresponds to
$L_{Rn}$, and remains constant from $\sim 10$ min after source insertion
onward.

#### Fitting range:

An important choice in the fitting process is the range of $r$ over which the
fit is performed. In the analysis of these experiments, two fit regions were
used: $0.5-1$ mm for the in-vivo set and $0.5-1.5$ mm for the ex-vivo one. The
reasoning behind these values is discussed in section 3.

#### Comparing between in-vivo and ex-vivo experiments:

As the activity profiles measured in the in-vivo sections showed significant
departure from the model for $r\gtrsim 1-1.5$ mm (while the ex-vivo profiles
followed the model more closely), a comparison parameter $\eta$ was defined
as:

$\eta(r)=\frac{\Gamma_{Measurement}(r)}{\Gamma_{Model}(r)}$ (3)

where $\Gamma_{Measurement}(r)$ is the measured activity profile of a given
tumor section, and $\Gamma_{Model}(r)$ is the profile calculated using
Equation (2) with $L_{Rn}$ and $A$ obtained from the fitting process applied
to the same section.

#### Validation using simulations:

Finally, simulations were performed with the aim of validating the approach
described here. This was done by generating synthetic activity plots using
known values of $L_{Rn}$, which were then “blindly” fitted with the
theoretical expression with the aim of extracting $L_{Rn}$. The results of
these calculations point to a convergence error of up to 3% between the
extracted value of $L_{Rn}$ using the fit process described here and the known
values of $L_{Rn}$ used for the simulations. The full details of the
validation process are given in Appendix B.

#### Approximating the DaRT source as a finite line:

This choice is based on the measured activity profiles. As is shown in section
3, many of the profiles exhibit behavior more akin to a line source than a
cylindrical one, i. e. the maximum of activity is at $r=0$ (or close to it)
rather then the expected $r=0.35$ mm (the source radius). It could be that in
these cases, the tumor tissue, which is still elastic after a short period of
irradiation, shrinks radially back to fill the hole left by the source. In
terms of the difference in expected activity profiles between the two cases
(line vs. cylindrical source), the only change is that in the cylindrical
case, the activity is “pushed” outward, but otherwise retains its spatial
form. This can be seen in Figure 3, which shows activity profiles for both
line and cylindrical sources for two values of $L_{Rn}$. This means that in
the fitting process, the extracted diffusion length is not affected by the
source radius (the free parameter $A$ does depend on it, but this parameter is
of no interest in this study).

Figure 3: Comparison between 10-mm-long line and cylindrical Alpha-DaRT
sources: 220Rn specific activity profiles, 30 min after source insertion, for
two values of $L_{Rn}$. The other model parameters are: $L_{Pb}=0.6$ mm,
$P_{leak}(Pb)=0.6$, $L_{Bi}=0.1L_{Pb}$, $\alpha_{Bi}=0$. The specific
activities are shown in the mid-plane of the source, which carries 3 $\upmu$Ci
224Ra, with $P_{des}(Rn)=0.45$ and $P_{des}^{eff}(Pb)=0.55$.

###  2.6 Uncertainty analysis

The uncertainties for extracted values of $L_{Rn}$ have three sources: (1)
Selection of the source location (around which the activity profiles are
calculated). Labeled $\Delta L_{Rn}^{loc}$, it was evaluated by comparing the
values of $L_{Rn}$ extracted when the location was selected manually and the
value, from the same section, when the location was selected by using the COG
method. This comparison was done for 10 sections (five from in-vivo tumors and
five from ex-vivo ones). The differences were then averaged, and the resulting
uncertainty calculated to be 6.4%. (2) Uncertainty introduced by the fitting
process (discussed in section 2.5 and shown in Appendix B, Figure 8), labeled
$\Delta L_{Rn}^{fit}$. Its value was taken as the maximal difference between
the fitted and known values, as shown in Figure 8b, resulting in an error of
3%. (3) Variation between different sections obtained from the same tumor,
labeled $\Delta L_{Rn}^{var}$. Its value, which was generally the dominant
uncertainly term, was taken as the standard deviation (SD) of the values
extracted from sections of the same tumor. This is the dominant uncertainty
and is generally of the order of $\sim$ 10%. Finally, all three were added in
quadrature to give the total uncertainty for each extracted $L_{Rn}$:

$\Delta L_{Rn}=\sqrt{\Big{(}\Delta L_{Rn}^{loc}\Big{)}^{2}+\Big{(}\Delta
L_{Rn}^{fit}\Big{)}^{2}+\Big{(}\Delta L_{Rn}^{var}\Big{)}^{2}}$ (4)

##  3 Results

Figures 4 and 5 show examples of measured sections from the ex-vivo and the
in-vivo tumors sets, respectively (four different tumors in each figure). The
left column shows the number of recorded alpha-particle hits per pixel
$N_{\alpha}(\textbf{r})$ for a single tumor section. In addition, it shows the
estimated location of the source (blue circle). In panel (a) of Figure 4, the
location estimated using the COG method is also shown (black dots),
emphasizing the need for a manual selection of the source location in some
cases. The right column shows the ring-averaged number of alpha-particle hits
per pixel $\langle N_{\alpha}(r)\rangle$ as a function of distance from the
source, as well as the fitted function used to extract $L_{Rn}$, Equation (2).
Although Equation (2) refers to the specific activity, rather than to $\langle
N_{\alpha}(r)\rangle$, the difference between the two is incorporated in the
scaling factor $A$ which is a free parameter in the fit. Each row shows data
from a different tumor: Figure 4, panels (a)-(b): SQ2, 0.23 g, (c)-(d): SQ2,
0.22 g, (e)-(f): SQ2, 0.17 g, (g)-(h): 4T1, 0.79 g. Figure 5, panels (a)-(b):
SQ2, 0.23 g, (c)-(d): PANC2, 0.71 g, (e)-(f): 4T1, 0.74 g, (g)-(h): 4T1, 0.36
g. The vertical bars in the right-side panels show the start and end of the
fit region: 0.5 - 1 mm for in-vivo tumors and 0.5 - 1.5 mm for ex-vivo ones.
The reason for the change of fit region is explained below. A good agreement
between the measured activity profiles shown here and the theoretical ones,
fitted using Equation (2), indicates that indeed the fitting method, even
though it is to an approximate source geometry (a superposition of point
sources), “captures” the leading effects of physical processes at short radial
distances (up to 1-1.5 mm from the source), namely, the diffusion of 220Rn.

Figure 4: Examples of measured sections from four different experiments of the
ex-vivo tumors set. The left column shows the number of recorded alpha-
particle hits per pixel for a single tumor section. In addition, it shows the
estimated location of the source (blue circle). In panel (a), the location
estimated using the COG method is also shown (black dots). The right column
shows the ring-averaged number of alpha-particle hits per pixel as a function
of distance from the source, as well as the fitted function used to extract
$L_{Rn}$, Equation (2). The vertical bars in the right-side panels show the
start and end of the fit region ($0.5-1.5$ mm). Each row shows data from a
different tumor; the specific cell line, tumor mass, and fitted $L_{Rn}$ are
stated in the right-side panels. Figure 5: Examples of measured sections from
four different experiments of the in-vivo tumors set. The left column shows
the number of recorded alpha-particle hits per pixel for a single tumor
section. In addition, it shows the estimated location of the source (blue
circle). The right column shows the ring-averaged number of alpha-particle
hits per pixel as a function of distance from the source, as well as the
fitted function used to extract $L_{Rn}$, Equation (2). The vertical bars in
the right-side panels show the start and end of the fit region ($0.5-1$ mm).
Each row shows data from a different tumor; the specific cell line, tumor
mass, and fitted $L_{Rn}$ are stated in the right-side panels.

Averaging over all diffusion length values extracted from the 14 in-vivo
tumors and comparing it to the 10 ex-vivo ones, there seems to be no
significant difference between the groups: 0.40 $\pm$ 0.08 mm for in-vivo
tumors and 0.39 $\pm$ 0.07 for ex-vivo ones. However, looking at the dose
profiles of the in-vivo tumors (Figure 5), it can be seen that at larger
radial distances the slope appears to change (decrease), an effect which is
not observed in ex-vivo tumor profiles (Figure 4). This is the reason for the
different fit regions between ex-vivo and in-vivo tumors.

We use $\eta(r)$, introduced in section 2.5 (equation 3), to quantify the
difference between the measured activity profile and the theoretical one.
Examples for calculated values of $\eta$ are shown in Figure 6. These were
calculated for the four sections shown in Figure 4a, 4g, 5a and 5e. The values
of $\eta$ in Figure 6 are plotted against $r^{*}$, defined as
$r^{*}=r/L_{Rn}$, with $L_{Rn}$ being the diffusion length extracted for the
specific section. It can be seen that indeed $\eta$ values increase noticeably
above 1 at distances of $r^{*}>2$ for the in-vivo SQ2 tumor and $r^{*}>4$ for
the in-vivo 4T1 tumor. Based on Figure 6, we chose $r^{*}=5$ as a fixed
reference point to evaluate the $\eta$, representing a point where significant
deviation from the model exists while ensuring $\eta$ is calculated inside the
tumor boundaries. The calculated $\eta(r^{*}=5)$ values are shown in Figure 7a
for the in-vivo tumors and 7b for the ex-vivo ones. The horizontal line is the
average value, and the dashed lines are $\pm$ one standard deviation. It can
be seen that the values measured in in-vivo tumors are significantly higher
then those of ex-vivo ones, suggesting that some contribution from non-
diffusive processes, e.g., vascular flow exists, enhancing the spread of
activity at large radial distances. While such processes are presently not
included in the DL model, they have an apparently negligible effect close to
the source, as indicated by the similar values of $L_{Rn}^{in-vivo}$ and
$L_{Rn}^{ex-vivo}$ and by the good agreement with the model over the fit
region.

The final results for the measured diffusion lengths are summarized in Figure
7c for the in-vivo tumors and 7d for the ex-vivo ones. As before, the
horizontal line is the average value, and the dashed lines represent one
standard deviation. For each tumor, at least two histological sections were
analyzed.

Figure 6: Examples of calculated $\eta$ values as a function of distance from
the source in two in-vivo and two ex-vivo tumors, expressed in units of
diffusion length, $r^{*}=r/L_{Rn}$. The experiment set, cell line, and tumor
mass are indicated in the figure. A horizontal line is added at $\eta=1$ for
reference. Figure 7: (a) Experimental results for $L_{Rn}$ measured in 14 in-
vivo tumors. The horizontal line represents the average value of $L_{Rn}$ and
the dotted lines are one standard deviation above and below that average. (b)
The same, for the 10 ex-vivo tumors. (c) Calculated $\eta$ values for the in-
vivo tumors, at distances equal to 5 times the measured diffusion length of
the respective tumor. As in (a), the horizontal line is the average and the
dotted lines are $\pm$ one standard deviation. (d) The same, for the ex-vivo
tumors.

##  4 Discussion

The similar values of $L_{Rn}$ extracted from ex-vivo and in-vivo tumors and
the good agreement with the model over the fit region $r=0.5-1$ mm or
$r=0.5-1.5$ mm suggest that the dominant process governing the spread of
activity over these regions is indeed diffusion (otherwise $L_{Rn}$ would be
larger in in-vivo tumors). A two-sample t-test comparing the in-vivo and ex-
vivo $L_{Rn}$ values, including all 24 diffusion lengths measured (14 in-vivo
and 10 ex-vivo) results in a p-value of 0.73, showing indeed no statistically
significant difference between the two groups. In addition, the average values
in both in-vivo and ex-vivo tumors are consistent with the reported diffusion
coefficient in water 45, with some cases higher than water - even for ex-vivo
tumors (we note that other studies have also claimed higher values for 220Rn
diffusion coefficient in water 48). Previously in Alpha-DaRT in-vivo studies,
such values (measured above 0.4 mm) were associated with a contribution to
diffusion from 212Pb, or $L_{Pb}$ 37 and led to the conclusion that, in most
tumors, the spread of activity is “lead-dominated”, i.e., the activity profile
is dictated by the diffusion length of 212Pb and the dose delivered $\gtrsim
1.5$ mm away from the source is the result of the single alpha particle
emitted by either 212Bi or 212Po. Now, the high values of $L_{Rn}$ suggest
that the Alpha-DaRT treatment may be, in fact, dominated by radon diffusion.
This will be further discussed in future publications. It should be emphasized
that the current $L_{Rn}$ results are reported from a relatively small
dataset. At this stage, it would be premature to conclude if $L_{Rn}$ varies
across tumor types, as the variability within a given tumor type seems to be
fairly large. Further measurements will be done to better quantify this.

It is important to address the enhanced spread of activity observed only in
in-vivo tumors (Figure 5 and 7c), which indicates that additional, beyond-
diffusion processes, are at play. Interstitial flows should not significantly
affect 220Rn spread: typical interstitial flow velocities are 0.1-1 $\upmu$m/s
49 so that even for an unrealistic uniform flow the convective length (the
mean distance traveled by 220Rn before its decay).
$L^{conv}_{Rn}=v\cdot\tau_{Rn}$ is $\sim 8-80$ $\upmu$m, much smaller than the
diffusion length. This process could affect small residual activities of
212Pb, where the much larger effective lifetime could lead to convective
spread on the scale of the diffusion length or higher. A different process, in
which vascularity could affect 220Rn spread, is a “hop-on/hop-off” mechanism,
where 220Rn atoms enter a blood vessel, travel with the blood flow for a short
distance, then leave the vessel and continue to spread via diffusion. A
mechanism of this sort could potentially lead to a higher-than-zero leakage
term for 220Rn (presumed negligible until now), but, as 220Rn is a noble gas,
the “transit time” inside blood vessels is expected to be short, preventing
significant leakage from the tumor. A recent study 50 measuring radon
solubility in various tissues also reported a lower solubility in diffusion-
only experiments and in-vivo samples where additional mechanism of
distribution via blood flow is present 51, 52. This, and other possible
mechanisms, will be further investigated in future studies.

Finally, although the aim of this study is to provide, for the first time, an
experimental value for $L_{Rn}$, we believe it is important to also relate
these results to the Alpha-DaRT treatment, as these measurement are intended,
ultimately, for use in treatment planning for Alpha DaRT. While the exact
process leading to the enhanced spread is still unknown to us, this spread
may, in principle, result in higher tumor doses than expected from the DL
model using a conservative fit over short radial distances. However, for this
to be the case the effect should be substantiated and shown to exist over a
few days after source insertion. For now, our recommendation is to adopt the
conservative approach and ignore it for tumor dosimetry. Such a conservative
choice may be needed regardless of the additional non-diffusive spread, to
compensate for unknown local variations in its magnitude. Even with such
effects included, the dose fall-off in healthy tissue is sufficiently rapid to
ensure sparing healthy tissue 2-3 mm away from the outermost source in a
lattice. This will be shown in a separate publication where measurements in
healthy pig tissues are discussed.

##  5 Conclusions

In this work we have presented the first measured values of the diffusion
length of 220Rn, $L_{Rn}$, in both in-vivo and ex-vivo mice-borne tumors of
varying types and masses. The measurement procedure, developed here for the
first time, allows measurement of $L_{Rn}$ despite 220Rn’s short half-life.
The values measured for $L_{Rn}$ lie in the range of $0.25-0.6$ mm, which
exceeds the previously assumed limit of 0.4 mm. This allows the scenario,
presumed unlikely until now, that the dominant contributor to the spread of
activity, in some tumors, is 220Rn, rather than 212Pb. This fact significantly
increases the calculated dose profiles for an Alpha-DaRT treatment planning,
easing the constraints on both source activities and lattice spacing. It was
further shown that the measured activity profiles exhibit a pure-diffusion-
based spread near the source surface ($r<1$ mm). However, an excess of
activity was consistently observed in in-vivo tumors far from the source,
which cannot be explained by the current model. This will be the target of
future studies.

## Acknowledgements

The SQ2 (squamous cell carcinoma), 4T1 (triple-negative breast cancer), PANC2
(pancreatic duct adenocarcinoma) and MC38 (colon carcinoma) cell lines were
provided by Prof. Yona Keisari (Sackler School of Medicine, Tel-Aviv
University, Israel). The C4-2 (prostate carcinoma) cell line was provided by
Dr. Martina Benešová (DKFZ-Heidelberg, German Cancer Research Center
Foundation, Heidelberg, Germany).

## Conflict of Interests

This work was partly funded by Alpha Tau Medical Ltd (ATM). ATM was not
involved in the study design and analysis, the writing of this article, or the
decision to submit it for publication. L.A. and T.C. are minor shareholders of
ATM. The scholarship of G.H. is partially paid by ATM and that of M.D. was
fully paid by ATM through a research agreement with Ben-Gurion University of
the Negev. The salary of I.L. is partially paid by ATM and that of M.V. is
fully paid by ATM. The scholarship and salary of N.W. are partially paid by
ATM. L.A. and T.C. are co-inventors of several Alpha-DaRT-related patents and
patent applications. G.H., M.D., M.V., and I.L. are co-inventors of pending
patent applications on Alpha-DaRT dose calculations.

## Data availability

The experimental data is available upon request to the corresponding author.

##  Appendix A The iQID autoradiography system

#### General description

The activity present inside a tumor section was recorded using the ionizing-
radiation Quantum Imaging Detector (iQID) autoradiography system 46. This
system consists of a scintillation material (a thin sheet of ZnS:Ag), an image
intensifier, a series of optical lenses and a fast CMOS camera connected to a
computer. During measurement, the tumor sections are placed on the
scintillation foil. When an alpha particle hits the ZnS:Ag layer, a flash of
light is created and amplified by the image intensifier (beta electron events
are excluded by setting a high threshold on the light intensity). The light is
then focused on the CMOS panel. Each flash of light recorded by the CMOS is
referred to as a “cluster”. The imaged clusters are continuously sent to the
computer, where each cluster’s central peak of intensity (i.e., its center-of-
gravity, COG) is found. The cluster COG serves to estimate the emission point
of the alpha particle. Given the 10 $\upmu$m thickness of the tumor section,
the accuracy in estimating the emission point is $\sim 20$ $\upmu$m and, since
these are estimated on an event-by-event basis, there is no need to perform
image deconvolution on the data. The iQID is usually operated at a refresh
rate such that there is no overlap of clusters in a frame. In this study, it
was operated at a refresh rate of 25 Hz. Finally, each cluster’s COG
coordinates in the $xy$ plane are saved in a list-mode file, along with its
time-stamp.

#### iQID components

The scintillation material used in this study is a 25 $\upmu$m layer of ZnS:Ag
deposited on a 100 $\upmu$m layer of transparent polyester plastic sheet
(Eljen technologies, EJ-440-100). The ZnS density is 3.25 $\pm$ 0.25 mg/cm2.
The iQID system used in this study is a commercial system from QScint Imaging
Solutions, with a 50-mm-diameter active area and pixel resolution of 24.15
$\upmu$m. The device’s efficiency was measured to be $0.46\pm 0.04$ (coming
from a one-sided measurement of the sample activity).

#### Efficiency calibration

The iQID’s alpha particle detection efficiency was calculated using two 212Pb
calibration samples. The samples were prepared by the following procedure. A
small cylindrical plastic container (28.5 mm in diameter and 3 mm in height)
is placed inside a low-pressure chamber ($<0.1$ mbar) with an open 228Th
surface source at the bottom of the container and an aluminum sheet glued to
its top. 224Ra ions recoiling from the 228Th source are implanted into the
aluminum sheet. After several days of collecting 224Ra ions, the activity on
the Al sheet approaches secular equilibrium with the parent 228Th source. In
order to prepare a calibration sample, the 224Ra-implanted container top is
removed and placed on top of another container, with a plastic sheet at the
bottom which will become the calibration sample. The 224Ra-implanted Al sheet,
similarly to what happens inside a tumor, releases its daughter atoms by
recoil into the container and the calibration sample at the bottom. The
exposure duration determines the calibration sample activity. Once this stage
is finished, the calibration sample is taken to a NaI(Tl) gamma counter 53 to
determine its 212Pb activity. Finally, the sample is taken to the iQID system,
where the alpha particles emitted from 212Bi and 212Po are measured. In order
to compare the activity of 212Pb with that of 212Bi and 212Po, the calibration
sample is set to “rest” for a few hours after collection, to ensure it reaches
secular equilibrium. The two calibration sample activities used here were set
to $\sim 10$ and 20 Bq, and the resulting efficiency of the iQID was measured
to be $0.46\pm 0.04$.

##  Appendix B Fit methodology validation

The validity of the $L_{Rn}$ extraction process presented here was tested
using the DART2D code 38, which allows simulating the procedure of these
experiments. These simulations produced artificial activity profiles for
varying values of source dwell times and diffusion lengths, which were then
blindly fitted using the process described in section 2.5 to extract an
“unknown” value of $L_{Rn}$, labeled $L_{fitted}$. These extracted values were
then compared to the known parameters used in the simulation.

The artificial activity profile were calculated as follows. DART2D 38 was used
to solve the time-dependent DL model equations for 220Rn, 212Pb and 212Bi for
a series of source dwell times ranging from minutes to days. For each case,
the system was allowed to evolve uninterrupted from source insertion to the
prescribed dwell time. The source was then “switched off” and the diffusion
coefficients of 220Rn, 212Pb and 212Bi were set to zero (simulating negligible
diffusion in the frozen—and then fixated—tumor tissue). The simulation
continued further under these conditions, with the dynamics reduced to that of
the local decay chain 220Rn $\rightarrow^{212}$Pb $\rightarrow^{212}$Bi. To
simulate the autoradiography stage, the local number of 212Bi decays was
integrated at the source mid-plane, for a duration consistent with the start
and end times of the actual measurement (typically starting 3 hours after
tumor removal and lasting $\sim 24$ hours). The recorded 212Bi decays were
then used to calculate the activity as a function of distance from the source
(the activity profile) in the source mid-plane.

Figure 8a shows the results (fitted diffusion length vs. dwell time) obtained
for a case where $L_{Rn}=0.3$ mm, $L_{Pb}=0.6$ mm and the other model
parameters are as before. The results are plotted for two different radial fit
regions mentioned above. In both cases, the fit converged to the simulated
value of $L_{Rn}$ for dwell times of several dozen minutes. On the other hand,
when the dwell time is long (days), the fit converges to a value which is
within $\sim 10-25\%$ of the dominant diffusion length—here $L_{Pb}$—with an
over-estimation than decreases when the fit is applied over a wider radial
region. The accuracy of convergence for various values of $L_{Rn}$ was
calculated as the relative error between the fitted value, $L_{fitted}$ and
the one used for the calculation, $L_{0}$. The results, shown in figure 8b for
$L_{Pb}=0.6$ mm and a source dwell time of 30 minutes, point to a convergence
error of up to 3%. Based on these results, the experimental dwell time was set
to 30 minutes, as a reasonable compromise allowing, on the one hand,
sufficient time for the fit to approach $L_{Rn}$, but at the same time
minimizing the buildup of 212Pb. We note that when the calculation was
repeated for other values of $L_{Pb}$, namely $L_{Pb}=0.3$ mm and $L_{Pb}=0.1$
mm, both for $L_{Rn}=0.3$ mm, the 3% convergence limit still held. Finally, we
note that the offset along $z$ relative to the source midplane has a
negligible effect ($<1\%$ change in convergence error), as long as the
distance from the source edges is larger then $\sim$1 mm.

Figure 8: (a) Fitted values of $L$ depending on the source dwell time for two
different radial regions: $0.5<r<1$ mm (blue circles) and $0.5<r<1.5$ mm (red
squares). The diffusion lengths used for the simulated activity profiles are
$L_{Rn}=0.3$ mm, $L_{Pb}=0.6$ mm. The source dwell time is 30 minutes and the
leakage probability is $P_{leak}(Pb)=0.6$. The other model parameters are the
same as before. (b) Relative error in the fitting process as a function of the
fitted diffusion length value, for the two radial fitting ranges used in the
data analysis.

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