#  Explaining Answers with Entailment Trees

Bhavana Dalvi*1, Peter Jansen*2, Oyvind Tafjord1, Zhengnan Xie2,
Hannah Smith2, Leighanna Pipatanangkura2, Peter Clark1
1 Allen Institute for AI, Seattle, WA
2 University of Arizona, Tucson, AZ
* equal contribution
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Our goal, in the context of open-domain textual question-answering (QA), is to
explain answers by not just listing supporting textual evidence
(“rationales”), but also showing how such evidence leads to the answer in a
systematic way. If this could be done, new opportunities for understanding and
debugging the system’s reasoning would become possible. Our approach is to
generate explanations in the form of entailment trees, namely a tree of
entailment steps from facts that are known, through intermediate conclusions,
to the final answer. To train a model with this skill, we created
EntailmentBank, the first dataset to contain multistep entailment trees. At
each node in the tree (typically) two or more facts compose together to
produce a new conclusion. Given a hypothesis (question + answer), we define
three increasingly difficult explanation tasks: generate a valid entailment
tree given (a) all relevant sentences (the leaves of the gold entailment
tree), (b) all relevant and some irrelevant sentences, or (c) a corpus. We
show that a strong language model only partially solves these tasks, and
identify several new directions to improve performance. This work is
significant as it provides a new type of dataset (multistep entailments) and
baselines, offering a new avenue for the community to generate richer, more
systematic explanations.111 EntailmentBank is available at
https://allenai.org/data/entailmentbank

##  1 Introduction

Figure 1: Given a hypothesis (green, summarizing a question+answer pair), and
some partially relevant text (or a corpus), our goal is to generate an
entailment tree, including intermediate nodes (blue), showing how the
hypothesis follows from the text/corpus.

Explanation remains a formidable challenge in AI. While today’s explanation
systems are good at listing supporting evidence (“rationales”) for an answer
DeYoung et al. (2019), they rarely explain the chain of reasoning from the
evidence to the answer, i.e., how the answer follows, given the evidence – the
goal of this work. Without this, it is hard to fully understand a system’s
response and/or pinpoint the source of errors if its conclusions are wrong.
Conversely, if a system could support its answers with a chain of reasoning,
new opportunities arise for interactively teaching the machine by debugging
its mistakes.

Our approach is to generate explanations in the form of multistep entailment
trees, such as shown in Figure 1, made up of individual textual entailment
(TE) steps Dagan et al. (2013). Unfortunately, although there are several
single-step entailment datasets available Bentivogli et al. (2011); Bowman et
al. (2015) no dataset of multistep entailments exists, and so a significant
contribution of this paper is the construction of such a dataset, called
EntailmentBank. EntailmentBank contains 1,513 multistep entailment trees for
accompanying QA pairs, constructed using expert annotators, and is the first
dataset of its kind. We then define three increasingly difficult explanation
tasks over this dataset, namely: generate a valid entailment tree for a given
QA pair given (a) all relevant sentences (the leaves of the gold entailment
tree), (b) all relevant and some distractor sentences, or (c) a full corpus.

Property$\downarrow$, Dataset$\rightarrow$ | WorldTree V2 | eQASC | R4C | StrategyQA  | EntailmentBank
---|---|---|---|---|---
Semantics of Inference  | (informal) | 1-Step Entailment | (informal) | Deduction  | Entailment Tree
Average Facts per Inference  | 6 | 2 | 2 | 3  | 7
Average Edges per Inference  |  9‡ | 1 |  2‡ | 2  | 6
Granularity of Inference  | Fine | Coarse | Coarse | Coarse  | Fine
Explicit Ordering of Inference  | No | No | No | Yes  | Yes
Authoring Method  | Expert | Crowd | Crowd | Crowd  | Expert

Table 1: A comparison of EntailmentBank with other similar datasets. In
general, EntailmentBank contains larger inference problems, at a finer level
of granularity than existing datasets, while being the only dataset to include
multi-step entailments that make the reasoning steps explicit. ‡ WT2 and R4C
explanations are implied (unannotated) graphs based on overlapping words or
entities – values here are inferred by constructing graphs based on lexical
overlap.

Our focus here is on generating the derivation (line of reasoning) showing how
the evidence leads to the answer, rather than the pragmatics of deciding which
parts of that to then show the user. This allows us to separate two (typically
confounded) explanation requirements, namely correctness (of the derivation)
from utility (of the pragmatics), allowing us to evaluate derivations with a
more objective measure (correctness). Our derivations also set the stage for
future research on the pragmatics of what to show the user Miller (2019).

We also define and train a generative model for this task, adapting techniques
used previously for generating deductive proofs Tafjord et al. (2020). We find
the model only partially solves the dataset, posing a challenge to the
community. Our contributions are thus:

  * •

A formulation of explanation as multistep textual entailment.

  * •

EntailmentBank, the first dataset of multistep entailment trees for QA, to
support entailment-based explanation. Each tree contains an average of 6.6
nodes and 2.7 entailment steps, with the full dataset of 1,513 trees including
a range of small and large multi-step entailment problems.

  * •

Baseline results using a state-of-the-art, generative model, showing that
reasonable trees (42% with zero errors) can be generated in a restricted
setting (all leaf sentences provided), but that the full task (tree from a
corpus) remains hard. We include a detailed analysis identifying several
avenues for future improvement.

This work is significant as it provides a new avenue, with a dataset and
baseline models, for the community to generate richer, more systematic
explanations.

##  2 Related Work

In the context of QA, there are multiple notions of explanation/justification,
including showing an authoritative, answer-bearing sentence Perez et al.
(2019), an attention map over a passage Seo et al. (2016), a synthesized
phrase connecting question and answer Rajani et al. (2019), or the syntactic
pattern used to locate the answer Ye et al. (2020); Hancock et al. (2018).
These methods are primarily designed for answers to “lookup” questions, to
explain where and how an answer was found in a corpus.

For questions requiring inference, the focus of this paper, an explanation is
sometimes taken as the chain of steps (typically sentences) leading to an
answer. Because of the difficulty of generating detailed decomposition steps,
existing datasets typically include one or more simplifications that
ultimately limit their utility (see Table 1 for a comparison). For example,
HotpotQA’s support task goes partway towards this goal by asking for answer-
supporting sentences Yang et al. (2018) or triples (in the derivative R4C
dataset Inoue et al. (2020)), but neither dataset describes how these
sentences combine to arrive at the answer. Similarly, WorldTree V2 Xie et al.
(2020) contains supporting sentences but no information about how they
combine. More recently, StrategyQA goes further by including derivations for
its answers, but using deductive steps and at a coarse granularity Geva et al.
(2021). Similarly, eQASC Jhamtani and Clark (2020) includes reasoning
annotations but limited to a single entailment step. In this work we
generalize to tasks requiring multi-step entailment trees. Table 1
illustrating these comparisons in more detail.

Our dataset also differs from human-authored explanation datasets, e.g.,
e-SNLI Camburu et al. (2018), and CoS-E Rajani et al. (2019). These datasets
were primarily designed for (explanation) language generation (without an
associated explanation semantics), while our goal is to assemble explanations
from an authoritative corpus so that they have credible provenance.

Recent work in deductive reasoning has shown that transformers can generate
formal proofs with high reliability, both in a formal setting Polu and
Sutskever (2020); Wang and Deng (2020) and with rules expressed in natural
language Saha et al. (2020). Inspired by this, we apply similar ideas to
generating entailment trees, in particular leveraging the generative
techniques used in the ProofWriter system Tafjord et al. (2020) (Section 5).

Finally, our task extends textual entailment (TE) in two ways. First, as well
as asking if text $T$ entails hypothesis $H$, we wish to know which part(s) of
$T$ entail $H$. Second, ours is the first dataset containing multistep
entailment trees, in contrast to earlier TE datasets where typically a single
sentence entails $H$ through (typically) paraphrasing Bentivogli et al.
(2011); Bar-Haim et al. (2014); Bowman et al. (2015).

Figure 2: The web-based authoring tool developed to enable authoring
entailment trees. (top) The question and a human-readable version of the semi-
structured explanation are provided to the user. (bottom) The semi-structured
explanation, including the entailment tree, as currently authored by the user.
Nodes (facts) can be dragged-and-dropped to change their ordering. White nodes
represent facts from the corpus, while orange nodes were authored by the user.
(right) A shortlist (or pool) of top-ranked relevant facts from the corpus
that the user can choose to drag-and-drop into the explanation.

##  3 The EntailmentBank Dataset

Figure 3: Two example medium-complexity entailment trees, paired with their
questions. The root nodes of each tree (hypotheses) are denoted by H (green),
and intermediate conclusions are blue. (top) An entailment tree with 4
entailment steps and 10 nodes describing the reasoning required to determine
why an astronaut requires oxygen in spacesuit backpacks. (bottom) An
entailment tree with 3 entailment steps and 7 nodes describing the reasoning
required to answer a question about the similarity of two concepts
(evaporation and condensation).

###  3.1 Overview

EntailmentBank contains two parts: 1,513 entailment trees, each tree showing
how a question-answer pair (QA) is entailed from a small number of relevant
sentences (e.g., Figure 1); and a general corpus $C$, containing those and
other sentences of domain-specific and general knowledge relevant to the QA
domain. We use these two parts shortly to define a simpler task (generate the
tree given the leaf sentences, without/with distractors) and a harder task
(generate the tree from the corpus). While EntailmentBank is geared towards
the science domain, the construction techniques could also be applied to
construct entailment trees in other domains.

EntailmentBank uses multiple-choice questions (and the correct answer option)
from the ARC dataset of grade-school science questions Clark et al. (2018),
and a corpus of science- and general knowledge derived from WorldTree V2 Xie
et al. (2020); Jansen et al. (2018). WorldTree includes a knowledge-base of
science-relevant knowledge, expressed as a collection of tables, but is easily
converted to a corpus of sentences for EntailmentBank as each table row is (by
design) readable as a fluent English sentence. WorldTree was created with ARC
questions in mind, making it an ideal source for EntailmentBank’s corpus.

###  3.2 Guidelines

Expert graduate and undergraduate annotators were trained to construct
entailment trees for QA pairs, given a small number of potentially relevant
sentences for each QA pair (drawn from the WorldTree corpus). Specifically,
they were trained to author trees:

  * •

where each step is an entailment (a conclusion that “a person would typically
infer” Dagan et al. (2013)), i.e., the knowledge expressed in each node
reasonably follows from the content of its immediate children.

  * •

at a fine-grained granularity, where each step encodes a single inference,
e.g., making a single taxonomic inference, conjoining two facts, or applying a
single rule in the corpus.

  * •

that are explicit, with the informal goal of including all the knowledge that
a young child would need to answer the question.

  * •

that are compositional, where more complex conclusions can be drawn from
simpler leaf facts.

  * •

that are accurate, concluding (a declarative version of) the QA pair of
interest.

####  3.2.1 Tool and Authoring Procedure

Constructing detailed entailment trees meeting the above desiderata is
challenging, even for expert annotators, and in pilot experiments we found
that authoring trees directly, from scratch, was extremely difficult. To make
authoring easier, we designed a web-based graphical drag-and-drop authoring
tool (see Figure 2) that allows explanation authors to construct, iterate, and
review explanations quickly.222The EntailmentBank authoring tool was
implemented as a Javascript browser client and npm back-end, and is released
as open source at https://github.com/cognitiveailab/entailmentbanktools/ .

For each question, the tool presents the user with a pool of top-ranked
relevant facts from the corpus that might be relevant to building an
explanation. To assist the tree construction process, the user first populates
an “explanatory worksheet”, identifying facts from the pool in a small number
of specific categories (e.g., “core facts”, “grounding facts”) that they
anticipate will be included in the tree. From this worksheet, the user then
begins constructing the entailment tree – typically starting at the bottom-
most leaf nodes, authoring intermediate conclusions from them, then
progressively working on higher levels of the tree until they author a
conclusion that directly answers the question, and all branches of the tree
meet detail and decomposition requirements. If the user requires a fact not
present in the pool of provided facts, e.g., a missing science fact or a
question-specific statement, the user can quickly compose their own facts and
add these to the tree. Once completed, the individual entailment steps are
then separately reviewed by a different author for quality and suggested
edits. In total, this process takes an average of approximately 20 minutes per
question. Two example trees authored using this process are shown in Figure 3.

####  3.2.2 Identifying Relevant Facts

To identify the pool of relevant facts to show the annotator for each
question, we could simply perform information retrieval over the corpus, using
the QA pair as the query. However, for this dataset, we are able to do better
than this: First, we train two “relevant sentence” classifiers (using BERT
Devlin et al. (2019) and RoBERTa Liu et al. (2019) respectively) using
additional WorldTree annotations.333 WorldTree includes annotations about
which WorldTree table rows are relevant to which questions, i.e., which rows
are supporting evidence (“rationales”) for which question. Although these
rationales do not identify all relevant sentences, they can be used as distant
supervision (along with random negative facts drawn from the corpus) to train
a “relevant sentence” classifier. Then, for each question, both models
exhaustively score each fact in the corpus, and the top 20 facts from each are
retrieved, reranked using Tensorflow-Ranking-BERT Han et al. (2020), and
presented as a ranked list to the entailment tree annotator based on their
final scores.

###  3.3 Overall Dataset

Due to the large time investment required to generate detailed entailment
trees, we author trees for 1,513 randomly selected questions, which include a
total of 3,968 discrete entailment steps. Overall, approximately 600 (paid)
work hours were used to build the dataset.

Figure 4: Histograms of entailment tree complexity, measured on the training set. (top) Proportion of trees with a given number of nodes (facts). (bottom) Proportion of trees with a given number of entailment steps. The average entailment tree contains 6.6 nodes (facts) across 2.7 entailment steps.  | Train | Dev | Test | All
---|---|---|---|---
Questions | 1079 | 163 | 271 | 1513
Entailment reasoning steps | 2857 | 408 | 703 | 3968
Table 2: Summary statistics for the dataset splits.

Summary statistics for the train, development, and test sets are shown in
Table 2. On average, each entailment tree includes 6.6 nodes across 2.7
entailment steps, where each entailment step typically involves 3 facts (two
leaves, that combine to entail a conclusion). Figure 4 shows histograms of
entailment tree size (measured in terms of the number of nodes, and number of
entailment steps). EntailmentBank includes a diverse range of problem sizes,
with approximately half (49%) of entailment trees representing short
entailment problems with one or two entailment steps (typically composed of
3-5 nodes), while the remaining 51% of trees contain 3-16 entailment steps (up
to 27 nodes).

Inference Type | Prop. |  | Example Entailment
---|---|---|---
Substitution | 42% | $s_{1}$ | when a light wave hits a reflective object, the light wave will be reflected
|  | $s_{2}$ | a mirror is a kind of reflective object
|  | $int$ | when a light wave hits a mirror, the light wave will be reflected
Inference from Rule | 33% | $s_{1}$ | if two species have similar characteristics, they may share a common ancestor
|  | $s_{2}$ | rhinoceroses and horses have similar characteristics
|  | $int$ | rhinoceroses and horses might share a common ancestor
Further Specification or | 15% | $s_{1}$ | an animal requires warmth for survival as the season changes to winter
Conjunction |  | $s_{2}$ | thick fur can be used for keeping warm
|  | $int$ | thick fur can be used for keeping warm as the season changes to winter
Infer Class from Properties | 4% | $s_{1}$ |  A compound is made of two or more elements chemically combined
|  | $s_{2}$ |  sodium chloride is made of two elements chemically combined
|  | $int$ |  sodium chloride is a kind of compound
Property Inheritance | 4% | $s_{1}$ |  an animal’s shell is usually hard
|  | $s_{2}$ |  something hard can be used for protection
|  | $int$ |  an animal’s shell is usually hard for protection
Sequential Inference | 3% | $s_{1}$ | In molecular biology, translation follows transcription
|  | $s_{2}$ |  transcription is when genetic information flows from DNA to RNA
|  | $s_{3}$ |  translation is when genetic information flows from RNA to proteins
|  | $int$ | In molecular biology, genetic information flows from DNA to RNA to proteins
Table 3: The prevalence of 6 common reasoning methods required to solve
individual entailment tree steps, sampled from 100 random entailment steps in
the training corpus. Discrete entailment steps in EntailmentBank require
diverse forms of reasoning to solve, from forms of taxonomic or merynomic
chaining (substitution) to application of domain-specific rules. Here, $s_{n}$
denotes input sentences, while $int$ denotes entailed conclusions
(intermediate nodes in the trees).

###  3.4 Dataset Analysis

To understand the entailment challenges in EntailmentBank, we analyzed 100
randomly sampled entailment steps from trees in the training set. We
identified 6 common high-level categories of inference, shown in Table 3.
Substitution types refer to entailments that require a model to perform
taxonomic, merynomic, or other forms of chaining that substitute one entity
for another in one of the input sentences. Inference from Rules entailments
require the application of a specific rule, specified as one of the input
sentences, to the other input sentence. Our analysis suggests that
approximately one-third (33%) of all entailments require the application of
domain-specific rules to complete. Further Specification or Conjunction
entailments require a model to combine the details of both input facts into a
single output fact. Less frequent types require inferring an object’s class
from it’s properties, inheriting properties of objects, or determining orders
for sequential reasoning. As a whole, this analysis shows diverse forms of
reasoning are required to successfully complete the entailment steps in
EntailmentBank.

##  4 Task Definitions

Because producing correct entailment trees from a corpus is challenging, we
define three tasks of increasing difficulty (named no-distractor, distractor,
and full-corpus, respectively) that simplify the retrieval and evaluation
problems inherent in the task. The inputs to all three are a hypothesis $H$,
namely a declarative form of a question + answer ($QA$),444 For convenience we
provide both $H$ and $QA$ as inputs, although in principle $H$ may be
generated from $QA$ automatically, e.g., using the QA2D model Demszky et al.
(2018) and some sentences $S$ expressing (both relevant and irrelevant)
knowledge about the world. The desired output is a valid entailment tree $T$
where the leaves are sentences selected from $S$, the intermediate nodes inti
are intermediate conclusions (new sentences, not part of the input), and the
root node (conclusion) is the hypothesis $H$. $T$ is valid if every node
$n_{i}$ in the tree is entailed by its children. Examples are shown in Figures
1 and 3.

As an approximation to make automated evaluation feasible, we ensure that $S$
includes all the leaf sentences $S_{gold}$ that are in the gold entailment
tree $T_{gold}$, and treat $T_{gold}$ (+ valid reorderings) as the only valid
entailment tree constructable from that input. This allows us to check
validity by comparing the generated tree with $T_{gold}$. This approximation
is reasonable for tasks 1 and 2 below, because their limited input makes it
unlikely that an alternative valid tree is constructable from the input. For
task 3, though, to avoid alternative valid trees being buildable from the
input corpus, we remove the few sentences similar to $S_{gold}$ from the
corpus on a per-question basis. Although these steps are not fool-proof, they
do allow entailment tree validity to be reasonably approximated by comparison
with $T_{gold}$, a critical methodological requirement for automatic
evaluation.

The three tasks, in increasing order of difficulty, are as follows:

  * Task 1 (no-distractor):

Inputs = $H$ \+ $QA$ \+ leaf sentences $S_{gold}$

  * Task 2 (distractor):

Inputs = $H$ \+ $QA$ \+ leaf sentences $S_{gold}$ \+ 20-30 distractor
sentences

  * Task 3 (full-corpus):

Inputs = $H$ \+ $QA$ \+ a corpus $C$ (that includes $S_{gold}$ and excludes
other sentences similar to $S_{gold}$)

Task 3 (full-corpus) represents the full task where $C$ is large, potentially
requiring one or more information retrieval (IR) steps to construct a valid
tree. In principle, $C$ is simply the WorldTree corpus of general facts.
However, in practice there are two question-specific deviations from this
ideal. First, annotators sometimes included non-WorldTree facts in the gold
entailment tree, either to express question-specific scenario facts (e.g., “A
ball rolls down a hill.”) or additional world knowledge missing in WorldTree.
To accommodate this, these extra facts are added to $C$ on a per-question
basis. Second, as noted earlier, we remove corpus sentences similar to
$S_{gold}$ to discourage alternative valid trees being found, so comparison
with $T_{gold}$ is a reasonable evaluation.

The desired output in all cases is a valid entailment tree $T$, approximated
as being the gold entailment tree $T_{gold}$ (+ valid reorderings).

##  5 Model

Inspired by the “All-at-once” sequence-to-sequence model in the ProofWriter
system Tafjord et al. (2020), we train a T5-based generative model for our
task, called EntailmentWriter.

###  5.1 Entailment Tree Encoding

We encode entailment trees as a linear structure that can be output by a
generative model. To do this, the input sentences $S$ are labeled with
identifiers (sent1, sent2, …), and the hypothesis $H$ is labeled with the
special identifier ‘hypot’ (Figure 1). All nodes in the output tree are then
identifiers: sent* for leaf nodes, int* for internal nodes, and ‘hypot’ for
the conclusion (root node). As the int* nodes denote new sentences (not in the
input), we include those sentences in the output immediately after their int*
identifier is first introduced.

When linearizing the tree, we start from leaf facts and work towards proving
the root of the tree (hypot). We use the symbol “&” to denote “and”, and
“-$>$” to denote “entails”. Thus the depth 2 entailment tree in Figure 1 would
be encoded as: [vskip=0mm,leftmargin=2mm] sent2 & sent5 -$>$ int1: Eruptions
block sunlight ; sent4 & int1 -$>$ hypot Note here that the new sentence
“Eruptions block sunlight” that intermediate node int1 denotes is explicitly
part of the to-be-generated output. The task for the models is to output valid
entailment trees encoded in this way, given the input.

###  5.2 Model Details

The EntailmentWriter model is built on top of the text-to-text pretrained T5
transformer Raffel et al. (2020), where the inputs are as described in Section
4 for Task 1 (no-distractor) and Task 2 (distractor). For Task 3 (full-
corpus), the corpus exceeds T5’s token limit, so we add a retrieval step of 30
sentences from the corpus $C$ using the hypothesis $H$ as query. The retrieved
sentences are then labeled with identifiers and input to the model. This may
mean that only a subset of the required sentences $S_{gold}$ is recalled and
provided to the model, hence the generated entailment tree will necessarily be
incomplete in such situations, given this strategy. (Note that other
strategies - for future work - could avoid this problem, e.g., using multiple
retrieval steps).

The output of the model is the full predicted entailment tree encoded as
described earlier (Section 5.1).

We fine-tune the models on the training set using the default hyperparameters
(including the Adafactor optimizer) in the T5
library.555https://github.com/google-research/text-to-text-transfer-
transformer We use the largest T5-11B model, fine-tuned for 20k steps (batch
size 8), selecting the checkpoint with highest validation score.

| Entailment Tree Scoring
---|---
|  | Leaves |  | Steps |  | Intermediates |  | Overall
|  | F1 | AllCorrect |  | F1 | AllCorrect |  | BLEURT | AllCorrect |  | AllCorrect
Task 1 (no-distractor) |  | 99.3 | 93.4 |  | 60.0 | 48.7 |  | 0.74 | 58.7 |  | 42.4
Task 2 (distractor) |  | 87.9 | 47.6 |  | 47.0 | 33.2 |  | 0.68 | 56.1 |  | 29.5
Task 3 (full-corpus) |  |  |  |  |  |  |  |  |  |  |
(Lower bound) |  | 34.9 | 2.2 |  |  0.3 | 0.0 |  | 0.00 | 21.0 |  |  0.0
(Upper bound, Task 2) |  | 87.9 | 47.6 |  | 47.0 | 33.2 |  | 0.68 | 56.1 |  | 29.5
Table 4: Baseline scores of the generated entailment trees from
EntailmentWriter, along four different dimensions (test set). F1/BLEURT scores
measure predicted/gold overlap, while AllCorrect scores 1 when all the
predictions are correct for a tree, 0 otherwise.

##  6 Experiments

###  6.1 Evaluation Metrics

We score the generated entailment trees along four dimensions: Do they use the
right input sentences (for leaf nodes of the tree)? Is the tree structurally
correct? Are the generated conclusions valid? Is the overall entailment tree
correct?

We approach evaluating entailment trees as a two step problem. First, nodes in
the predicted tree $T_{pred}$ are aligned with nodes in gold tree $T_{gold}$.
Then, we compare both the structure (edges) of the gold and predicted trees,
as well as the content of the gold and predicted nodes. To perform the
alignment, accounting for partial overlap and different valid ways of writing
the same entailment tree, we proceed as follows:

  * •

First, for each intermediate conclusion $int_{pred}$ in $T_{pred}$, and
$int_{gold}$ in $T_{gold}$, we gather their ancestor leaf sentences.

  * •

Then, we align each intermediate node $int_{pred}$ to the first $int_{gold}$
for which the Jaccard similarity of their respective ancestor sentences is
maximum. For any $int_{pred}$ with zero Jaccard similarity to all gold nodes
$int_{gold}$, it is aligned to a dummy gold node with a blank conclusion.

Once aligned, the aligned tree $T_{pred}^{{}^{\prime}}$ is scored against gold
tree $T_{gold}$ using the metrics below. The F1/BLEURT metrics score elements
of the tree (micro-averaging the results), while “AllCorrect” checks if all
the elements are correct (1=yes, 0=no), i.e., the predicted tree is perfect
along the dimension being considered. Our metrics are:

  * Leaf Nodes (F1, AllCorrect):

Does the predicted tree use the correct leaf sentences? We compute an F1 score
by comparing leaf sentences $S_{pred}$ to $S_{gold}$. The “AllCorrect” score
is 1 if the entire set of nodes is identified correctly (F1=1.0), 0 otherwise.

  * Steps (F1, AllCorrect):

Does the tree have the correct entailment step structure? Here we compute F1
and “AllCorrect” scores in a way similar to leaf nodes by comparing the set of
entailment steps in the aligned proof $T_{pred}^{{}^{\prime}}$ with
$T_{gold}$.

  * Intermediates (BLEURT, AllCorrect):

Are the synthesized intermediate nodes $Int_{pred}^{{}^{\prime}}$ AllCorrect.
For this metric, we use the state-of-the-art BLEURT metric for evaluating
generations, using the off-the-shelf BLEURT-Large-512 model.666We fine-tuned
and evaluated the BLEURT metric further using 300 hand-labeled examples,
however it performs similarly to the vanilla model. Sellam et al. (2020). Our
analysis based on 300 hand-labelled examples from the development set suggests
that it correlates well with human ratings ($correlation=0.67$,
$sensitivity=0.88$, $specificity=0.80$). Further, we define generation
correctness as 1 if an aligned pair of $int_{pred}^{{}^{\prime}}$,
$int_{gold}$ gives $BLEURT>0.28$,777 The BLEURT threshold was picked using a
subset of 300 manually labeled pairs. When we test this threshold on the rest
of the labeled pairs we get a high (89%) F1 score, indicating the threshold is
reasonable. 0 otherwise. Note that a correct intermediate doesn’t necessarily
imply a correct entailment, as the ancestor nodes may be wrong.

Finally, “Intermediates AllCorrect” per question is 1 if all the generated
intermediates in the tree are correct, 0 otherwise.

  * Overall Proof (AllCorrect):

We define the overall “AllCorrect” score for a generated proof as 1 only if
all of the leaves, steps, and intermediates are all correct, i.e., the tree
completely matches $T_{gold}$. Otherwise it scores 0. This is a strict metric:
any error in the generated tree will result in a score of 0.

###  6.2 Main Results

Table 4 shows performance of the EntailmentWriter model on each of the three
subtasks. In the easiest setting, Task 1 (no-distractor), where only the gold
leaves are provided as input, the model correctly structures these into gold
entailment steps in 49% of cases, and generates 58% of intermediate
conclusions correctly. In total, EntailmentWriter is able to generate gold
entailment trees for only 42% of questions in this easiest setting,
highlighting the difficulty of this task.

Task 2 (distractor) increases the difficulty of identifying relevant leaves by
adding distractors to the input gold leaf sentences until a total of 30
sentences are supplied as input to the model888On average, each input in the
distractor setting has 25 distractor sentences. In this setting, the model is
able to correctly identify gold leaf nodes only about half the time (47%),
which significantly reduces overall task performance from 42% (no-distractor)
to 30% (distractor). In spite of this reduction in overall task performance
due to the loss of correct leaf nodes, correctness at generating intermediate
conclusions for each entailment step remains nearly the same as in the no-
distractor setting at 56%, suggesting that when EntailmentWriter has
meaningful input facts, it is able to make use of them equally well.

In Task 3 (full-corpus), the input is the entire WorldTree corpus, augmented
with additional question-specific and general facts used in gold entailment
trees but not part of WorldTree. Given this input is too large for a
transformer, a retrieval step is needed. Here, we provide upper- and lower-
bound results corresponding to perfect and weak retrieval results
respectively.

  * Upper bound:

As the Task 2 input contains exactly the required sentences $S_{gold}$ (+
distractors), it is equivalent to the perfect retrieval scenario. Hence Task 2
results can be used as an upper bound for Task 3.

  * Lower bound:

We run our Task 2 model using sentences retrieved from WorldTree alone (using
Tensorflow-Ranking-BERT retrieval, Section 3.2.2). As this retrieval does not
see any required non-WorldTree facts, the model is necessarily disadvantaged,
hence we use this score as a lower bound on Task 3.

As Table 4 shows, the lower bound result is poor, even struggling to select
the right leaf nodes (F1 = 34%), and not generating any perfectly correct
trees. Although this model is disadvantaged, it suggests that performing well
on the full task is difficult.

###  6.3 Error Analysis

To better understand the challenges in generating multistep entailment trees,
we performed error analyses both at the level of individual entailment steps,
as well as failures associated with generating full trees.

####  6.3.1 Individual Entailment Steps

We first analyze cases where the model is failing at individual entailment
reasoning steps. For this we sampled 100 entailment steps from the development
set where the predicted tree had overall AllCorrect = 0 based on the automated
evaluation metrics, i.e., the automated metrics detected at least one error
somewhere in the tree. Manually evaluating the sampled entailments, we obtain
the statistics shown in Table 5. The analysis suggests that even though these
trees are invalid overall, 30% of their individual entailment steps are valid.

AllCorrect | % Entailment steps
---|---
Incorrect | 57%
Nearly correct | 13%
Correct | 30%
Table 5: Human evaluation of individual entailment steps in (development set)
trees automatically scored as (overall) not correct, i.e., contain some error.
Although the overall trees are invalid, 30% of their contained entailment
steps are correct.

In cases where an entailment step was invalid, we identify three common kinds
of failure:

  * •

The entailed conclusion simply repeats one of the input sentences (41%): This
can be attributed to the fact that the model has seen a lot of instances in
the training data where the intermediate conclusions have high word overlap
with input sentences. In the future, we would like to explore a training
regime where the loss function encourages the model to add something novel
compared with the input sentences.

  * •

The entailed conclusion does not follow from input sentences (47%): In these
cases, the model is using knowledge unstated in the input for this particular
entailment step but present somewhere else in the input context. In future, we
would like to address this by exploring an iterative generation approach that
focuses on one entailment step at a time and that can distinguish good vs bad
entailment.

  * •

The entailed conclusion is correct, but either different from gold or
irrelevant to prove the hypothesis (12%).

Number of | Number of | Leaves | Steps | Intermediates | Overall
---|---|---|---|---|---
steps | questions | F1 | AllCorrect | F1 | AllCorrect | BLEURT | AllCorrect | AllCorrect
1 | 84 | 95.1 | 83.3 | 81.3 | 79.8 | 0.95 | 86.9 | 79.8
2 | 83 | 88.6 | 43.4 | 40.0 | 22.9 | 0.69 | 54.2 | 12.0
3 | 43 | 85.8 | 37.2 | 27.3 | 9.3 | 0.51 | 44.2 | 7.0
4 | 27 | 83.8 | 25.9 | 26.1 | 0.0 | 0.51 | 29.6 | 0.0
$\geq$5 | 34 | 74.2 | 0.0 | 20.8 | 0.0 | 0.32 | 20.6 | 0.0
Any | 271 | 87.9 | 47.6 | 47.0 | 33.2 | 0.68 | 56.1 | 29.5
Table 6: Results on Task 2 (distractor) broken down by the number of
entailment steps in the gold tree, indicating that scores drop rapidly as
trees get larger (more steps).

####  6.3.2 Failure Analysis of Generated Entailment Trees

We analyzed 50 generated entailment trees in the development set where the
automated metrics gave an “AllCorrect” score of 0, signifying there is (at
least) one error somewhere in the tree. We observe the following classes of
errors:999Note that some error classes are not mutually exclusive, so
proportions do not add to 100%

  * •

Incorrect leaves results in invalid/missing entailment steps (56%): For
example, for the question “Why do mosquitoes instinctively move towards carbon
dioxide…? A: It helps mosquitoes find food”, the predicted tree misses using
the critical input fact that “mosquitoes eat animal blood”, hence cannot infer
“animals are a source of food for mosquitoes”, hence cannot infer the
importance of moving towards carbon dioxide.

  * •

Correct leaves, but invalid steps (18%): For example, for a question asking
“Can a person see someone in a dark room? A: No”, the model selects the
correct leaf sentences but stitches them together in the wrong order. As a
result, an invalid intermediate conclusion is generated. Specifically, it
incorrectly tries to draw an entailment from “a person is in a dark room” and
“a person is looking into the dark room”, producing “the person outside can
see the person in the dark room”, an invalid step and one that directly
contradicts the target QA pair.

  * •

Correct steps, but incorrect intermediate conclusions (2%): For example, for a
question with H:“compression waves cause objects to move in the same direction
of the wave”, the model gets the proof structure right in terms of placement
of relevant leaves. But instead of concluding a gold intermediate conclusion
“$int1_{gold}$: longitudinal waves are also called compression waves” it
prematurely predicts the final conclusion H for the intermediate $int1_{pred}$
(and then re-predicts it in the final entailment step).

  * •

Disconnected trees (4%): We found 2 examples where the generated entailment
tree had intermediate conclusions that were not used later towards proving the
hypothesis. E.g. for question with H: “the plants in the gardens will receive
the most sunlight in summer to grow during the day”, the model accurately
compose leaf facts to reach the intermediate conclusion “the plants in the
gardens will receive sunlight to grow” but the predicted proof does not then
use it in later steps. One way to avoid this would be to apply structural
constraints on the output, enforcing a (single) tree structure Saha et al.
(2020).

  * •

Imperfect evaluation (24%): For these cases we found that the generated
entailment tree was correct but our metrics could not detect the equivalence
in the predicted vs gold tree. Failure cases included 1) alternate valid
entailment trees missing in our gold annotations 2) the generated proof is
finer-grained or coarse-grained with respect to the gold proof. An improved
evaluation metric would help alleviate these scoring discrepancies.

####  6.3.3 Performance wrt. Tree Size

Finally, we grouped the Task 2 results according to the size (number of steps)
in the gold tree. The results are shown in Table 6, and demonstrate that the
scores drop significantly as the number of steps in the gold proof increases.

##  7 Summary and Conclusion

Our goal is to enable machines to generate richer, more systematic
explanations. To this end, we have developed a novel formulation of
explanations as multistep entailment trees, and created EntailmentBank, the
first large-scale dataset of such trees.

We have also presented baseline results for automatically generating
entailment tree explanations for answers to science questions, trained on
EntailmentBank. These initial results suggest that such generation is possible
in restricted settings (leaf sentences are provided), but that the full task
(explanation from a corpus) remains challenging. As new mechanisms are
developed to guide and validate tree construction, we expect the quality of
the resulting explanations to improve significantly. If this were possible,
new opportunities for understanding and debugging a system’s reasoning would
arise, allowing users to not just receive answers, but also have meaningful
dialogs with the machine about those answers by exploring its line of
reasoning. EntailmentBank contributes to this direction, offering a new
resource for developing richer, more systematic explanations. EntailmentBank
is available at https://allenai.org/data/entailmentbank

## References

  * Bar-Haim et al. (2014) Roy Bar-Haim, I. Dagan, and Idan Szpektor. 2014.  Benchmarking applied semantic inference: The pascal recognising textual entailment challenges.  In _Language, Culture, Computation_.
  * Bentivogli et al. (2011) L. Bentivogli, Peter Clark, I. Dagan, and Danilo Giampiccolo. 2011.  The seventh pascal recognizing textual entailment challenge.  _Theory and Applications of Categories_.
  * Bowman et al. (2015) Samuel R. Bowman, Gabor Angeli, Christopher Potts, and Christopher D. Manning. 2015\.  A large annotated corpus for learning natural language inference.  In _EMNLP_.
  * Camburu et al. (2018) Oana-Maria Camburu, Tim Rocktäschel, Thomas Lukasiewicz, and Phil Blunsom. 2018\.  e-SNLI: Natural language inference with natural language explanations.  In _NeurIPS_.
  * Clark et al. (2018) Peter Clark, Isaac Cowhey, Oren Etzioni, Tushar Khot, Ashish Sabharwal, Carissa Schoenick, and Oyvind Tafjord. 2018.  Think you have solved question answering? Try ARC, the AI2 reasoning challenge.  _ArXiv_ , abs/1803.05457.
  * Dagan et al. (2013) Ido Dagan, Dan Roth, Mark Sammons, and Fabio Massimo Zanzotto. 2013.  _Recognizing Textual Entailment: Models and Applications_.  Morgan and Claypool.
  * Demszky et al. (2018) Dorottya Demszky, Kelvin Guu, and Percy Liang. 2018.  Transforming question answering datasets into natural language inference datasets.  _ArXiv_ , abs/1809.02922.
  * Devlin et al. (2019) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019.  BERT: pre-training of deep bidirectional transformers for language understanding.  In _NAACL-HIT_.
  * DeYoung et al. (2019) Jay DeYoung, Sarthak Jain, Nazneen Rajani, E. Lehman, Caiming Xiong, R. Socher, and Byron C. Wallace. 2019.  ERASER: A benchmark to evaluate rationalized NLP models.  In _ACL_.
  * Geva et al. (2021) Mor Geva, Daniel Khashabi, Elad Segal, Tushar Khot, D. Roth, and Jonathan Berant. 2021.  Did Aristotle use a laptop? A question answering benchmark with implicit reasoning strategies.  _ArXiv_ , abs/2101.02235.
  * Han et al. (2020) Shuguang Han, Xuanhui Wang, Mike Bendersky, and Marc Najork. 2020.  Learning-to-rank with BERT in TF-ranking.  _ArXiv_ , abs/2004.08476.
  * Hancock et al. (2018) Braden Hancock, Paroma Varma, Stephanie Wang, Martin Bringmann, Percy Liang, and Christopher Ré. 2018.  Training classifiers with natural language explanations.
  * Inoue et al. (2020) N. Inoue, Pontus Stenetorp, and Kentaro Inui. 2020.  R4C: A benchmark for evaluating RC systems to get the right answer for the right reason.  In _ACL_.
  * Jansen et al. (2018) Peter A. Jansen, Elizabeth Wainwright, Steven Marmorstein, and Clayton T. Morrison. 2018.  WorldTree: A corpus of explanation graphs for elementary science questions supporting multi-hop inference.  In _LREC_.
  * Jhamtani and Clark (2020) Harsh Jhamtani and P. Clark. 2020.  Learning to explain: Datasets and models for identifying valid reasoning chains in multihop question-answering.  In _EMNLP_.
  * Liu et al. (2019) Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019.  RoBERTa: A robustly optimized BERT pretraining approach.  _ArXiv_ , abs/1907.11692.
  * Miller (2019) T. Miller. 2019.  Explanation in artificial intelligence: Insights from the social sciences.  _Artif. Intell._ , 267:1–38.
  * Perez et al. (2019) Ethan Perez, Siddharth Karamcheti, Rob Fergus, Jason Weston, Douwe Kiela, and Kyunghyun Cho. 2019.  Finding generalizable evidence by learning to convince Q&A models.  In _EMNLP_.
  * Polu and Sutskever (2020) Stanislas Polu and Ilya Sutskever. 2020.  Generative language modeling for automated theorem proving.  _ArXiv_ , abs/2009.03393.
  * Raffel et al. (2020) Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, M. Matena, Yanqi Zhou, W. Li, and Peter J. Liu. 2020.  Exploring the limits of transfer learning with a unified text-to-text transformer.  _J. Mach. Learn. Res._ , 21:140:1–140:67.
  * Rajani et al. (2019) Nazneen Rajani, B. McCann, Caiming Xiong, and R. Socher. 2019.  Explain yourself! Leveraging language models for commonsense reasoning.  In _ACL_.
  * Saha et al. (2020) Swarnadeep Saha, Sayan Ghosh, Shashank Srivastava, and Mohit Bansal. 2020.  PRover: Proof generation for interpretable reasoning over rules.  In _EMNLP_.
  * Sellam et al. (2020) Thibault Sellam, Dipanjan Das, and Ankur P Parikh. 2020.  Bleurt: Learning robust metrics for text generation.  _ArXiv_ , abs/2004.04696.
  * Seo et al. (2016) Minjoon Seo, Aniruddha Kembhavi, Ali Farhadi, and Hannaneh Hajishirzi. 2016.  Bidirectional attention flow for machine comprehension.  In _ICLR_.
  * Tafjord et al. (2020) Oyvind Tafjord, B. D. Mishra, and P. Clark. 2020.  ProofWriter: Generating implications, proofs, and abductive statements over natural language.  _ArXiv_ , abs/2012.13048.
  * Wang and Deng (2020) Ming-Zhe Wang and Jun Deng. 2020.  Learning to prove theorems by learning to generate theorems.  In _NeurIPS_.
  * Xie et al. (2020) Zhengnan Xie, Sebastian Thiem, Jaycie Martin, Elizabeth Wainwright, Steven Marmorstein, and Peter Jansen. 2020.  WorldTree V2: A corpus of science-domain structured explanations and inference patterns supporting multi-hop inference.  In _LREC_.
  * Yang et al. (2018) Zhilin Yang, Peng Qi, Saizheng Zhang, Yoshua Bengio, William W Cohen, Ruslan Salakhutdinov, and Christopher D Manning. 2018.  HotpotQA: A dataset for diverse, explainable multi-hop question answering.  _ArXiv_ , abs/1809.09600.
  * Ye et al. (2020) Qinyuan Ye, Xiaozhen Huang, and Xiang Ren. 2020.  Teaching machine comprehension with compositional explanations.  _ArXiv_ , abs/2005.00806.


# Upper Bound on Implantable Antennas
Considering Ohmic Loss

Jakub Liska, Mingxiang Gao, Lukas Jelinek, Erik R. Algarp, Anja K. Skrivervik,
Miloslav Capek Manuscript received ; revised .This work was supported by the
Czech Science Foundation under project No. 21-19025M, and by the Grant Agency
of the Czech Technical University in Prague under project No.
SGS22/162/OHK3/3T/13.J. Liska, L. Jelinek and M. Capek are with the Czech
Technical University in Prague, Prague, Czech Republic (e-mails: {jakub.liska;
lukas.jelinek; miloslav.capek}@fel.cvut.cz).M. Gao, E. Algarp and A. K.
Skrivervik are with École Polytechnique Fédérale de Lausanne, Lausanne,
Switzerland (e-mails: {mingxiang.gao; erik.algarp;
anja.skrivervik}@epfl.ch).Color versions of one or more of the figures in this
paper are available online at http://ieeexplore.ieee.org.

###### Abstract

A hybrid approach for the computation of performance limitations for implanted
antennas is presented, taking radiation efficiency as the optimized metric.
Ohmic loss in the antenna and surrounding tissue are both considered. The
full-wave interaction of all parts of the system is taken into account. The
method is thoroughly tested on a realistic implanted antenna design that is
treated both experimentally and as a model in a commercial electromagnetic
solver. A great agreement is reported. In addition, the fundamental bounds on
radiation efficiency are compared to the performance of a loop and a dipole
antenna showing the importance of various loss mechanisms during the designs.
The trade-off between tissue loss and antenna ohmic loss indicates critical
points in which the optimal solution drastically changes and in which the real
designs should change their topology.

###### Index Terms:

Implanted biomedical devices, hybrid solution methods, antenna efficiency,
fundamental bounds, tissue loss, ohmic losses, numeric methods, quadratic
programming.

##  I Introduction

Telemedicine has experienced a significant boom during the past few decades.
An important enabler of this progress is wireless communication and wireless
power transfer for medical implants via radiofrequency fields, which renders
the antenna a crucial element. Laws of physics, however, bound the performance
of antennas, and the knowledge of these limits is essential to improve the
development of telemedicine devices.

Performance limitations on antennas are of key interest to the antenna
designers [1, 2, 3, 4, 5, 6] even outside telemedicine and their determination
has evolved to a discipline on its own. For antennas radiating into free
space, they can nowadays be evaluated for arbitrary metrics and shapes [7, 8,
9].

Nevertheless, the aspects of wireless communication with implanted devices
[10, 11] differ significantly from antennas operating in free space. Of
particular interest for this paper are physical bounds on the maximum power
density of EM (EM) waves reaching free space from implanted antennas [12, 13,
14, 15, 16]. Generally, the size of an implanted antenna is typically a few
millimeters, and the operating frequency ranges from hundreds of megahertz to
a few gigahertz [17]. This indicates that implanted antennas are usually
electrically small. Contrary to conventional antennas, the implant is firstly
surrounded by a small volume of ideally lossless medium representing the
implant encapsulation. Beyond this, the EM waves radiated by the implant must
pass through a lossy tissue before escaping to free space.

Until recently, fundamental bounds for implanted antennas have only been
treated considering elementary sources [12] with no considerations on true
physical dimensions. Finite geometry, cylindrical shell, was considered in
[16] and compared to elementary infinitesimally small sources studied before.
An initial attempt to find the radiation efficiency limitation of an implanted
antenna with finite conductivity and arbitrary shape of the current supporting
region was presented in [18, 19]. It was followed by a feasibility study of
performance limitations on implanted antennas considering arbitrarily-shaped
and lossy current-supporting regions [20].

A popular methodology [7, 21] for evaluating fundamental bounds is based on
optimal current density, integral equation formulation, and the employment of
the method of moments. This presents a major obstacle for implanted devices
which demand the interaction of an electrically small antenna and a
potentially electrically large body111The radius of the smallest sphere
circumscribing the body is much larger than a typical operating wavelength in
the free space.. A direct approach using the volumetric meshing of
inhomogeneous tissues would be unbearably computationally demanding [22]. To
overcome the computation difficulties, a hybrid method [23] is used in this
work, which consists of a combination of two frequency domain schemes: the MoM
(MoM) formulation of the electric field integral equation is used for the
antenna and its closest neighborhood, while a vector spherical wave expansion
(VSWE) is applied to numerically evaluate the interaction of the EM field with
the host body [24]. This allows treating this complex problem with relatively
few unknowns [24]. The hybrid method requires a spherical boundary on which
the two underlying numerical methods, MoM and VSWE, are coupled. This boundary
is defined naturally as a spherical encapsulation of the current support, see
Fig. 1.

Figure 1: Within the used hybrid method, the model is separated into regions
treated by different numerical methods: MoM description of the current
supporting region denoted as $\varOmega_{Z}$, and VSWE description of the host
body denoted as $\varOmega_{T}$. The spherical boundary around the antenna
(implanted capsule) separates the two methods and provides their
interconnection.

Though the hybrid method is known, it remains to determine how fundamental
bounds can be formulated in this framework. The basic idea was introduced in
[18, 20, 19] and is developed in this paper. It also employs some previously
developed optimization schemes [21, 25]. Similarly to antennas in the free
space [7, 21], the computation of fundamental bounds [21, 25] is based on
convex optimization, particularly quadratically constrained quadratic program
(QCQP) [26, 21]. The approach is applied to a basic scenario with a spherical
single-layered phantom and commonly used medical frequencies of 403 MHz and
2.45 GHz. Results are verified by measurement and by simulations using
independent commercially available solvers: CST [27] (CST) simulation based on
FDTD and FEKO [28] (FEKO) simulation based on MoM.

This paper is organized as follows. Section II presents the used model and the
basis of the numerical tools used later. Their description is necessary to
understand the following sections. Section III stresses the importance of
finite metal conductivity of the current supporting region and consequences.
The method verification is done in Section IV, which also presents results of
simulations in commercially available solvers and measurement results. Further
investigation and discussion in the setup for verification are given in
section V.

##  II Computational Tools

Modeling the wireless link between an implanted antenna and an external node
is a challenging problem, resolved here using the advantages of two distinct
approaches, MoM and VSWE, which are combined within the hybrid method. MoM
with piece-wise basis functions [29] is applied to characterize the antenna or
current supporting region in a great level of spatial details. Conversely,
VSWE serves as a set of entire-domain basis functions to determine the
influence of the surrounding tissues. Computational complexity is
significantly reduced [30, 31] as compared to the case, in which the entire
setup is solved just with a single method with entire-domain discretization.

Within the quadratic representation of power-like quantities in MoM and VSWE,
the physical limitations are readily evaluated as in previous studies related
to MoM [7, 25, 21].The evaluation yields the value of the fundamental bound on
a chosen metric and an associated optimal current distribution, which may
serve as an initial inspiration for the antenna designers.

###  II-A Method of Moments and T-matrix Hybrid

MoM and T-matrix methods are described in [24] and only the essential steps to
hybridize them are presented here. As mentioned above, the method applies MoM
for the antenna description and VSWE with T-matrix for the surrounding
tissues.

####  II-A1 Method of Moments

Within MoM, the current supporting region222Support for optimized current
density in the sense of fundamental bounds defined in Section II-B or simply
the antenna metalization in the case of a design., $\varOmega_{Z}$ region in
Figs. 1 and 2, is discretized in a set of elementary cells together with
current density, which is approximated by a sum of weighted basis functions
$\left\\{\boldsymbol{\psi}_{n}\right\\}_{n=1}^{N}$ with expansion coefficients
$I_{n}$,

$\boldsymbol{J}(\boldsymbol{r})\approx\sum\limits_{n=1}^{N}I_{n}\boldsymbol{\psi}_{n}(\boldsymbol{r}).$
(1)

Substitution of (1) in the appropriate EFIE (EFIE) and the use of Galerkin
method [32] results in a system of linear equations

$\left(\mathbf{Z}_{0}+\mathbf{Z}_{\rho}\right)\mathbf{I}=\mathbf{V}_{\mathrm{i}}$
(2)

where $\mathbf{Z}_{0}$ is the free-space impedance matrix, $\mathbf{Z}_{\rho}$
is the material impedance matrix333The free-space impedance matrix
$\mathbf{Z}_{0}$ gives the properties of the current as it flows in the free
space, meaning the radiation (real part) and reactive power (imaginary part).
The material impedance matrix $\mathbf{Z}_{\rho}$ gives the properties of the
current flowing in the chosen material, meaning the losses (real part) and
reactive power of the material (imaginary part)., $\mathbf{V}_{\mathrm{i}}$ is
the excitation vector, and $\mathbf{I}$ is the unknown vector of current
expansion coefficients $I_{n}$. This allows for the characterization of the
antenna in the free space. It should also be noted that the addition of
dielectric inclusions or the use of non-spherical encapsulation can be
resolved by the use of a volumetric electric field integral equation in the
same way as described above.

Figure 2: Setup of the field and current expansion [18, 19]. Current density
in the current supporting region $\varOmega_{Z}$ is described via vector
$\mathbf{I}$, while VSWE of EM fields on the boundaries of the host body
$\varOmega_{T}$ is collected in vectors $\mathbf{a}_{1},\ \mathbf{f}_{1}$ and
$\mathbf{a}_{2},\ \mathbf{f}_{2}$ outside and inside the host body,
respectively. Spherical boundaries of these regions and the host body are
essential for the application of VSWE to describe the interaction of the
fields with the host body.

####  II-A2 Vector Spherical Wave Expansion

The idea of field expansion into a set of basis functions is also applied in
the T-matrix method. In this case, entire-domain basis functions are used, and
the electric and magnetic fields are expanded in vector spherical harmonics

$\displaystyle\boldsymbol{E}(\boldsymbol{r})$
$\displaystyle=k\sqrt{Z}\sum\limits_{\alpha}a_{\alpha}\mathbf{u}_{\alpha}^{(1)}(k\boldsymbol{r})+f_{\alpha}\mathbf{u}_{\alpha}^{(4)}(k\boldsymbol{r}),$
(3) $\displaystyle\boldsymbol{H}(\boldsymbol{r})$
$\displaystyle=\mathrm{j}\dfrac{k}{\sqrt{Z}}\sum\limits_{\alpha}a_{\alpha}\mathbf{u}_{\bar{\alpha}}^{(1)}(k\boldsymbol{r})+f_{\alpha}\mathbf{u}_{\bar{\alpha}}^{(4)}(k\boldsymbol{r}),$
(4)

where $k$ and $Z$ are wave number and background wave impedance, respectively,
$a_{\alpha},~{}f_{\alpha}$ are the expansion coefficients, and
$\mathbf{u}_{\alpha}^{(1)},~{}\mathbf{u}_{\alpha}^{(4)}$ are vector spherical
waves [24, Appendix A] of regular and outgoing type, respectively. To describe
the scenario of implanted antenna, the relation between the field outside the
body and in the implant (antenna) encapsulation is given in matrix form as

$\begin{bmatrix}\mathbf{f}_{1}\\\
\mathbf{a}_{2}\end{bmatrix}=\begin{bmatrix}\mathbf{T}&\mathbf{\Psi}\\\
\mathbf{\Psi}^{\mathrm{T}}&\mathbf{\Gamma}\end{bmatrix}\begin{bmatrix}\mathbf{a}_{1}\\\
\mathbf{f}_{2}\end{bmatrix},$ (5)

with $\mathbf{T},~{}\mathbf{\Gamma}$ being transmission matrices outside and
inside, respectively, matrices $\mathbf{\Psi},~{}\mathbf{\Psi}^{\mathrm{T}}$
accounting for field penetrating from inside to outside or vice-versa, and
$\mathbf{a},~{}\mathbf{f}$ gathering expansion coefficients. Sub-indices “1”
and “2” corresponds to expansion outside and inside, respectively; see Figs. 2
and 3.

Figure 3: Schematic diagram representing the matrix description of the host
body within VSWE.

In this paper, neither an incident field from infinity nor an external antenna
is considered, therefore, $\mathbf{a}_{1}=\mathbf{0}$ is used for simplicity
in further derivations.

In the case of a spherically layered host body, the interaction matrices in
(5) can be evaluated analytically [33], which is also done in this paper. A
representative example is a spherical encapsulation with an antenna centered
in a phantom of a human head. The human head can be sufficiently approximated
by a spherical multilayered host body [23], where tighter bounds on implanted
antennas, as compared to [12, 15], can be evaluated in terms of seconds using
the technique proposed here. The method can be easily adapted to other
scenarios like ingestible devices and is not restricted to medical telemetry,
but T-matrix has to be obtained numerically by some standard solver in the
case of a non-spherical model.

####  II-A3 Hybridization

The coupling of MoM and T-matrix method is done via projection matrix
$\mathbf{U}_{1}$

$U_{1}^{\alpha
n}=k\sqrt{Z}\left\langle\mathbf{u}_{\alpha}^{(1)},\boldsymbol{\psi}_{n}\right\rangle$
(6)

which maps vector spherical waves to basis functions from MoM as

$\displaystyle\mathbf{f}_{2}$ $\displaystyle=-\mathbf{U}_{1}\mathbf{I},$ (7)
$\displaystyle\mathbf{V}$
$\displaystyle=\mathbf{V}_{\mathrm{i}}+\mathbf{U}_{1}^{\mathrm{T}}\mathbf{a}_{2}.$
(8)

The above relations are written for a particular case of outgoing waves and
complete excitation within MoM, where $\mathbf{V}_{\mathrm{i}}$ is the direct
excitation in the current supporting region (for example delta-gap source).
This connection allows combining the methods from (2) and (5) into a hybrid
method

$\begin{bmatrix}\mathbf{Z}_{0}+\mathbf{Z}_{\rho}&-\mathbf{U}_{1}^{\mathrm{T}}&\mathbf{0}&\mathbf{0}\\\
-\mathbf{U}_{1}&\mathbf{0}&-\mathbf{1}&\mathbf{0}\\\
\mathbf{0}&-\mathbf{1}&\mathbf{\Gamma}&\mathbf{0}\\\
\mathbf{0}&\mathbf{0}&\mathbf{\Psi}&-\mathbf{1}\end{bmatrix}\begin{bmatrix}\mathbf{I}\\\
\mathbf{a}_{2}\\\ \mathbf{f}_{2}\\\
\mathbf{f}_{1}\end{bmatrix}=\begin{bmatrix}\mathbf{V}_{\mathrm{i}}\\\
\mathbf{0}\\\ \mathbf{0}\\\ \mathbf{0}\end{bmatrix}.$ (9)

###  II-B Upper Bound on Radiation Efficiency

Concerning implanted antennas, radiation efficiency is one of the most
important performance metrics. The performance is generally low due to the
lossy tissues surrounding the antenna. The previously studied limits [12, 14]
considered cycle-mean radiated power $P_{\mathrm{rad}}$ and cycle-mean power
absorbed by the lossy biological tissues $P_{\mathrm{tis}}$.

Within the hybrid method shown in the previous section II-A, ohmic losses in
the antenna can also be considered and separated from the total loss.
Similarly to [18, 20, 19], the upper bound on radiation is here formulated as
the optimal current density, which realizes the highest radiation efficiency.
The expansion coefficients $I_{n}$ are taken as the degrees of freedom in this
optimization problem. Explicitly, the radiated power $P_{\mathrm{rad}}$ over
the total power $P_{\mathrm{tot}}$ supplied to the system is maximized

$\eta_{\mathrm{rad}}^{\mathrm{ub}}=\max\limits_{\mathbf{I}}\eta_{\mathrm{rad}},$
(10)

where

$\eta_{\mathrm{rad}}=\dfrac{P_{\mathrm{rad}}}{P_{\mathrm{tot}}}=\dfrac{P_{\mathrm{rad}}}{P_{\mathrm{rad}}+P_{\mathrm{ant}}+P_{\mathrm{tis}}},$
(11)

with $P_{\mathrm{ant}}$ being the cycle-mean lost power in the current
supporting region $\varOmega_{Z}$ and $P_{\mathrm{tis}}$ being the cycle-mean
lost power in region $\varOmega_{T}$.

The individual power terms are defined within the hybrid method as follows

$\displaystyle P_{\mathrm{rad}}$
$\displaystyle=\dfrac{1}{2}\mathbf{f}_{1}^{\mathrm{H}}\mathbf{f}_{1},$ (12)
$\displaystyle P_{\mathrm{ant}}$
$\displaystyle=\dfrac{1}{2}\mathbf{I}^{\mathrm{H}}\mathrm{Re}\left[\mathbf{Z}_{\rho}\right]\mathbf{I},$
(13) $\displaystyle P_{\mathrm{tis}}$
$\displaystyle=\dfrac{1}{2}\left(\mathbf{f}_{2}^{\mathrm{H}}\mathbf{f}_{2}+\mathrm{Re}\left[\mathbf{a}_{2}^{\mathrm{H}}\mathbf{f}_{2}\right]\right)-P_{\mathrm{rad}}.$
(14)

Note that the first term on the right-hand side of (14) is the net cycle-mean
outward power flow at the inner boundary of $\varOmega_{T}$, see [24, Appendix
E] for derivation.

As mentioned above, the current density expansion coefficients $I_{n}$ are the
optimization variables, and the first row of system (9)
$\left(\mathbf{Z}_{0}+\mathbf{Z}_{\rho}\right)\mathbf{I}-\mathbf{U}_{1}^{\mathrm{T}}\mathbf{a}_{2}=\mathbf{V}_{\mathrm{i}}$
is therefore not imposed444Note that equation system (9) has a unique
solution, which binds the excitation vector with current vector. In order to
set up an upper limit to radiation efficiency of all antennas fitting the
prescribed current supporting region, this unicity must be relaxed. One of the
possible ways is to skip the first row of equation system (9), which gives the
necessary freedom to the current vector $\mathbf{I}$. But, particular lines in
the first row can be used as power constraints, to further tighten the bounds
and to impose constraints on input impedance, the shape of the feed, etc.
[34]. Optimization only relies on the boundary conditions between different
regions, which are represented by other rows. Specifically, the equation
system

$\begin{bmatrix}-\mathbf{U}_{1}&\mathbf{0}&-\mathbf{1}&\mathbf{0}\\\
\mathbf{0}&-\mathbf{1}&\mathbf{\Gamma}&\mathbf{0}\\\
\mathbf{0}&\mathbf{0}&\mathbf{\Psi}&-\mathbf{1}\end{bmatrix}\begin{bmatrix}\mathbf{I}\\\
\mathbf{a}_{2}\\\ \mathbf{f}_{2}\\\
\mathbf{f}_{1}\end{bmatrix}=\begin{bmatrix}\mathbf{0}\\\ \mathbf{0}\\\
\mathbf{0}\end{bmatrix}$ (15)

is used as an affine constraint in the optimization problem, while the current
density vector $\mathbf{I}$ is optimized so as to maximize radiation
efficiency.

As seen in (12) – (14), all the power terms are represented as quadratic
functions of expansion coefficients. Converting all terms (12) – (14) to the
functionals of vectors $\mathbf{I}$ via (15), optimization problem (10) is
transformed to quadratically constrained quadratic program (QCQP). Its
solution can be approached numerically [21], but in this particular case, it
can be solved more efficiently. After several algebraic manipulations and
substitutions from (15) to (12)–(14), the optimization problem (10) reduces to
GEP (GEP)

$\left(\mathbf{U}_{1}^{\mathrm{T}}\mathbf{\Psi}^{\mathrm{T}}\mathbf{\Psi}\mathbf{U}_{1}\right)\mathbf{I}=\eta\left(\mathrm{Re}\left[\mathbf{Z}_{0}+\mathbf{Z}_{\rho}\right]+\mathbf{U}_{1}^{\mathrm{T}}\mathbf{\Gamma}\mathbf{U}_{1}\right)\mathbf{I},$
(16)

where the left-hand-side matrix is proportional to the cycle-mean radiated
power (12) and the right-hand side to the cycle-mean total supplied power. The
maximum radiation efficiency $\eta_{\mathrm{rad}}^{\mathrm{ub}}$ is given by
the maximum eigenvalue $\eta$, and the optimal current density corresponds to
the associated eigenvector $\mathbf{I}$.

##  III Different Kind of Losses

Implanted antennas suffer from dissipation of power in two different regions:
the antenna itself and the host body. So far, only idealized EM dipole sources
were considered in [12, 16], concluding that elementary electric dipole
suffers more from tissue losses than the magnetic source, mainly due to the
coupling of the near field and the lossy tissues. In this section, we show
that the metal losses of a given current supporting region should also be
taken into account.

Let us consider two frequencies within standard medical bands:
$f_{1}=403\,\mathrm{MHz}$ and $f_{2}=2.45\,\mathrm{GHz}$. Further, let us
consider the following example: the current supporting region is a copper disc
with radius $r$ swept from 0.75 mm to 7.5 mm placed in a centered spherical
air bubble with radius $a=7.5\,\mathrm{mm}$ and implanted in spherical muscle
($\varepsilon_{\mathrm{r,m}}=57-36\mathrm{j}$ at frequency $f_{1}$ and
$\varepsilon_{\mathrm{r,m}}=53-13\mathrm{j}$ at frequency $f_{2}$) phantom
with radius $R=61.5\,\mathrm{mm}$. The current supporting region is supposed
to be made of copper with a thickness much higher than skin depth and,
therefore, can be modeled by a sheet of surface resistivity [35]
$R_{\mathrm{s}}=\sqrt{(2\pi f\mu_{0})/(2\sigma)}$, where conductivity
$\sigma=5.56\cdot 10^{7}\,\mathrm{S/m}$ and permeability of vacuum $\mu_{0}$
are assumed. The upper bound on radiation efficiency is computed for different
radii $r$ and is shown in Fig. 4.

Figure 4: Upper bound on radiation efficiency for frequencies $f_{1},f_{2}$
and radius of the disc normalized by the radius of the air bobble
$r/a\in[0.1,1]$. Performance bounds are evaluated for the copper disc as the
current support centered inside air bobble with radius $a=7.5\,\mathrm{mm}$,
which is placed in the center of the muscle-filled sphere with radius
$R=61.5\,\mathrm{mm}$.

Fig. 4 depicts, at each frequency, the optimal radiation efficiency as a
function of normalized radius $r/a$. This efficiency cannot be overcome by any
real antenna fitting the considered disc region. An interesting phenomenon is
an abrupt growth in radiation efficiency at frequency $f_{1}$ for support
sizes larger than $r/a=0.32$. For current supports up to a radius of
$r/a=0.5$, it is convenient to use frequency $f_{2}$ with stable performance
among different radii. In contrast, for larger sizes, the first frequency is
preferred.

A better physical insight into the aforementioned phenomenon is attained by
analyzing the power quantities in (12) – (14) normalized to the radiated power
$P_{\mathrm{rad}}$, i.e., dissipation factors of the current support/tissue

$\delta_{\mathrm{ant/tis}}=\dfrac{P_{\mathrm{ant/tis}}}{P_{\mathrm{rad}}},$
(17)

and the total dissipation factor

$\delta_{\mathrm{rad}}=\delta_{\mathrm{ant}}+\delta_{\mathrm{tis}}=\dfrac{1-\eta_{\mathrm{rad}}}{\eta_{\mathrm{rad}}}.$
(18)

The comparison of all dissipation factors is shown in Fig. 5.

At frequency $f_{1}$, the dissipation factors $\delta_{\mathrm{ant}}$ and
$\delta_{\mathrm{tis}}$ are discontinuous, with the critical point being the
radius $r/a=0.32$. For smaller sizes, tissue loss dominates, while the antenna
loss is comparable for larger sizes. At frequency $f_{2}$, all curves are
smooth, and antenna loss is negligible as compared to tissue loss.

Figure 5: Dissipation factors corresponding to the upper bound on radiation
efficiency shown in Fig. 4. The partial dissipation factors of the current
supporting region and the muscle host body are also shown. The radius of the
disc is, as in Fig. 4, normalized by the radius of the air bobble.

The behavior of power losses can be explained by observing the optimal current
density for the case of small and large current supporting regions. For the
smaller size, the optimal current densities are shown in the left panels of
Fig. 6.

(a)  (b)  (c)  (d)

Figure 6: Optimal current densities on a disc in a muscle host body
corresponding to data reported in Fig. 5 and Fig. 4: (top left)
$\mathrm{TM}_{1m}$-like for $r/a=0.1$ and $f_{1}=403\,\mathrm{MHz}$, (top
right) $\mathrm{TE}_{1m}$-like for $r/a=1$ and $f_{1}=403\,\mathrm{MHz}$,
(bottom left) $\mathrm{TM}_{1m}$-like for $r/a=0.1$ and
$f_{2}=2.45\,\mathrm{GHz}$, (bottom right) $\mathrm{TM}_{1m}$-like for $r/a=1$
and $f_{2}=2.45\,\mathrm{GHz}$.

The current densities are both $\mathrm{TM}_{1m}$-like555Current density is
denoted as $\mathrm{TE}_{1m}$-like (magnetic dipole), $\mathrm{TM}_{1m}$-like
(electric dipole), or $\mathrm{TM}_{2m}$-like (electric quadrupole) if its
projection to VSWE generates only $\mathrm{TE}_{1m}$ mode, $\mathrm{TM}_{1m}$
mode, or $\mathrm{TM}_{2m}$ mode, respectively. Subscript $m$ corresponds to
the azimuthal number and only reflects a rotation of a coordinate system.. For
the larger size of the current supporting region, the optimal current density
is, see the right panels in Fig. 6, $\mathrm{TE}_{1m}$-like at frequency
$f_{1}$ and is $\mathrm{TM}_{1m}$-like at frequency $f_{2}$. This explains the
discontinuity in dissipation factors at frequency $f_{1}$, where the optimal
current must dramatically switch its character to maximize radiation
efficiency. The reason for this is that $\mathrm{TE}_{1m}$-like current
couples less to the lossy dielectric host body [12] and therefore become
favorable when the boarders of the current supporting region come closer to
the tissue. While small-size $\mathrm{TE}_{1m}$-like currents can be seen as
an analogy to a short circuit [20]. At frequency $f_{2}$, this is not
necessary since the electrical size of the current support is big enough to
accommodate $\mathrm{TM}_{1m}$-like current with an optimal trade-off antenna
loss due to dense current in the center and tissue loss due to short distance
from the tissues. But, in general, antenna loss decreases with the increasing
$r$ for $\mathrm{TM}_{1m}$-like and $\mathrm{TE}_{1m}$-like currents because
more spreading of the current is allowed.

##  IV Experimental Verification

The following two sections show how to use the previously developed
methodology in the real design of an implanted antenna. This provides an
experimental validation of the method and also demonstrates some new features
that were not observed previously.

To allow for a simple experimental setup, the host body is simulated by a
glass bottle filled with distilled water, see Fig. 7 and Fig. 8. Distilled
water has well-determined EM properties
($\varepsilon_{\mathrm{r,w}}=78-12\mathrm{j}$ at frequency $f_{2}$) and
similarly well defined are EM properties of glass
($\varepsilon_{\mathrm{r,g}}\approx 6.7$). These values are also used by
commercial EM solver CST, which is employed as a virtual version of the
experiment. The outer radius of the bottle is $R_{1}=63.5\,\mathrm{mm}$ and
glass thickness is $t=2.0\,\mathrm{mm}$.

Figure 7: Measurement setup placed in the anechoic chamber. The ceramic-
encapsulated antenna is placed in the center of a spherical phantom filled
with distilled water.  Figure 8: Simplified scenario of the measurement: c –
ceramic, w – water, g – glass. The parameters are: spherical glass bottle
($\varepsilon_{\mathrm{r,g}}=6.7$) with an outer radius of
$R_{1}=63.5\,\mathrm{mm}$, a glass thickness $t=2\,\mathrm{mm}$, water radius
$R_{2}=R_{1}-t=61.5\,\mathrm{mm}$, ceramic encapsulation $a=7.5\,\mathrm{mm}$.
The true shape of the antenna ($\varOmega_{Z}$) is detailed in Fig. 9. In the
case of the fundamental bound, the current supporting region is a disc of
radius $r\in\left[0.75,7.5\right]\,\mathrm{mm}$.

The implanted antenna, see Fig. 9, is designed and measured for frequency
$f_{2}$. It consists of a $100$ µm thick polyimide disc
($\varepsilon_{\mathrm{r}}=3.5$) with $18$ µm thick copper cladding and two
ceramic hemispheres as its encapsulation. A meander dipole antenna is etched
in the copper. The antenna is connected to a copper jacketed semi-rigid cable
(EZ-34, EZ Form Cable) through a chip balun (2450BL15B050, Johanson
Technology) and a short coplanar stripline. A slot in the top ceramic
hemisphere is machined to accommodate the balun and the cable. The ceramic
hemispheres are made of $\mathrm{Al_{2}O_{3}}$ (alumina,
$\varepsilon_{\mathrm{r}}=12$ and $\mathrm{tan}\delta=0.0003$, CST model) that
is bio-compatible and also provides good matching of the antenna.

Figure 9: Photograph and design of the ceramic-encapsulated implanted antenna:
antenna encapsulated in the ceramic hemispheres (left top panel), decomposed
antenna and encapsulation (right top panel), visualization of the meander
dipole antenna including dimensions and interconnection with balun via
$3.2\,\mathrm{mm}$ long coplanar stripline with a strip width of
$0.18\,\mathrm{mm}$ and a gap width of $0.1\,\mathrm{mm}$ (left bottom panel),
encapsulation hemisphere’s dimensions (center and right bottom panel).

At first, the above-mentioned antenna setup is used to validate the hybrid
method used to define the fundamental bound. For this aim, the gain pattern of
the antenna was measured in an anechoic chamber, see Fig. 10 and appendix A
for details. Apart from the measurement and the hybrid method, the gain
patterns were also evaluated in commercial solvers: CST and FEKO. All
simulations omit the thin, flexible substrate and losses in the ceramic
encapsulation. All the gain patterns are depicted in Fig. 10 and show
excellent agreement. The maximum gain values $G(\pi/2,0)$ are
$-16.9\,\mathrm{dB}$ for simulated pattern by the hybrid method and using
Antenna Toolbox for Matlab [36] (AToM), $-16.9\,\mathrm{dB}$ in CST,
$-16.5\,\mathrm{dB}$ in FEKO, and $-17.6\,\mathrm{dB}$ for the measurement.
The computational requirements of the commercial solutions (using the FDTD
method in CST and the surface equivalence formulation in FEKO) were
considerable due to the use of high dielectric constants and overall large
electric size. The data presented in Fig. 10 demanded hours of computer time.
Compared to this, the hybrid method demanded only a few seconds and order in
magnitude less computer memory for the same task with the same discretization
of the dipole antenna. Even for the maximum possible meshing allowed by the
used computer, the results from the commercial solvers were still slightly
mesh dependent, and the hybrid method is therefore considered as more reliable
for this task.

Figure 10: Simulated and measured gain patterns’ cuts plotted in dB scale at
the frequency $f_{2}$ using the setup in Fig. 7. The cut plane corresponds to
$\theta=90\degree$. Polarization in the plane of the dipole antenna is
measured.

The radiation efficiency of the experimental setup, see Fig. 11, is also
compared with the performance of a loop and dipole antenna, which were made of
a copper strip of width $w=0.5\,\mathrm{mm}$ and fed by an ideal delta-gap
voltage source. The results once more show good agreement between several
numerical simulations, with only slight variations attributed to different
numerical schemes. Importantly, the figure also shows the fundamental bound on
radiation efficiency with a current supporting region in the form of a disc
accommodating all used antenna models.

Figure 11: The radiation efficiency of the meandered dipole antenna as
simulated with the hybrid method, CST, and FEKO at frequency $f_{2}$. The
performances of a simple circular loop and a straight dipole antenna are also
shown. The fundamental bound on the performance of a disc current supporting
region is the upper bound on the radiation efficiency of all considered
antenna designs. The dipole antenna, loop antenna, and fundamental bound
correspond to varying sizes of the current support, the size of which is
measured by the radius of the smallest circumscribing sphere $r$. The sizes
are normalized to the radius of the ceramic encapsulation, i.e., at $r/a=1$
the antenna fills the entire capsule.

The performance of the experimental setup lies only approximately
$1\,\mathrm{dB}$ below the fundamental bound. This means that for this
material composition, and shape of the design region (disc), not much can be
done to improve the design. The hybrid method shows that the interaction with
the host material is mostly dipolar and therefore does not depend much on the
detailed composition of the host body. It can be expected that this design
will similarly perform also in real-body conditions.

##  V Discussion

In order to understand the design principles of implanted antennas in more
detail, it is worth studying the simple cases of loop and dipole antennas in
Fig. 11. The dipole antenna has a positive trend in radiation efficiency with
increasing length, while the loop antenna achieves its maximum performance
around the radius of $r=0.6a$. These trends can be explained by separating
dissipation factors into three contributions as depicted in Fig. 12. With the
increasing length of the dipole antenna, the tissue dissipation factor is
constant, but on the contrary, the antenna dissipation factor declines, which
is inherited by the total dissipation factor. The antenna dissipation factor
of the circular loop antenna completely dominates for the small radius up to
$r/a=0.3$. Antenna dissipation factor decay is faster with the increasing
radius for the circular loop than for the straight dipole antenna due to the
increase of the impedance. However, with a circumscribing sphere radius of
$0.5a$, the circumference of the circular loop antenna is comparable to the
wavelength in the surrounding medium, and a further increase stops the
decrease of the antenna dissipation factor and increase of tissue dissipation
factor.

Figure 12: Dissipation factors of the meander dipole (black markers), a
circular loop (orange lines), and a straight dipole antenna (blue lines). The
same normalization as in Fig. 11 is used.

Due to high antenna losses, the performance gap (distance to fundamental
bound) for small electrical sizes is significant for straight dipole antennas
(being comparable to tissue losses) and even wider for loop antennas. To see
the reason for this behavior, it is instructive to plot how the dissipation
factors, see Fig. 13, corresponding to the fundamental bound, behave when
changing the size of the current supporting region.

Figure 13: Dissipation factors corresponding to the upper bound on radiation
efficiency for the water host body. The same normalization as in Fig. 11 is
used.

The total dissipation factor is approximately constant, and it is dominated by
the tissue dissipation factor. While the antenna dissipation factor is
discontinuous at $r/a\approx 0.35$. As can be seen in Fig. 14, for small
electric sizes, $\mathrm{TM}_{1m}$-like current realizes the optimal current
density. In contrast, $\mathrm{TM}_{2m}$-like current is optimal at higher
electric sizes. A switch between these two current profiles is represented by
an abrupt jump seen in Fig. 13. This critical point at $r/a\approx 0.35$
happens because of a slight improvement in radiation efficiency: a small drop
in tissue dissipation factor and an increase in antenna dissipation factor;
the radiation on an implanted antenna must always be seen as a trade-off
between the antenna and tissue dissipation factors. The
$\mathrm{TM}_{2m}$-like current shows that in certain scenarios, antennas
using higher-order modes can perform better than the ones using only the
dipole currents.

(a)  (b)  (c)  (d)

Figure 14: Optimal current densities on a disc as the current supporting
region in the water phantom at $f_{2}$. The corresponding radii are (from left
to right) $r/a=\\{0.1,0.4,0.8,1\\}$. The current densities are (from left to
right)
$\\{\mathrm{TM}_{1m},\mathrm{TM}_{2m},\mathrm{TM}_{2m},\mathrm{TM}_{2m}\\}$-like.

##  VI Conclusion

The use of the hybrid method for the computation of radiation efficiency
limitations for implanted antennas is shown in this paper. The hybrid method
not only allows for a dramatic reduction of computational demands (order in
magnitude faster than commercial solutions of the same setup) but also reveals
different contributions of losses in the problem of an antenna radiating
through lossy media. The technique is applied to a disc-like current
supporting region that is encapsulated in a dielectric and placed in the host
body, i.e., a setup able to accommodate many of the practical implanted
antenna designs. The paper shows how the upper bound on the radiation
efficiency of all these designs can simply be evaluated. Furthermore, the
paper reveals the importance of ohmic loss in the antenna proper compared to
tissue loss.

Varying the size of the current supporting region, the trade-off between ohmic
losses in the antenna and the tissue drastically changes the shape of the
optimal current density. The critical points were identified where the
$\mathrm{TE}_{1m}$-like current density performs better than a
$\mathrm{TM}_{1m}$-like current and vice-versa. This proves the importance of
the fundamental limitations on the performance of implanted antennas and the
usefulness to evaluate them for the scenario before starting the design to
know the bounds.

The method is verified by measurement as well as computation in commercial
solvers, with the example being a meander dipole with a balun immersed in a
phantom filled with distilled water. Great agreement is achieved for the
results from the hybrid method, commercial simulators, and measurement. A
detailed study of the fundamental bound on radiation efficiency in this
scenario has also been done, concluding that ohmic loss in an implanted
antenna must be considered in the design.

##  Appendix A Measurement

The implanted meander dipole antenna within a glass bottle filled with
distilled water gain pattern is measured at a distance of 2.1 m from the
reference antenna (a quad-ridge horn antenna, QH400, MVG Industries), see Fig.
7. The meander dipole antenna is oriented along the $y$ axis.

The reflection and transmission coefficients were measured from 2 GHz to 3
GHz, see Fig. 15, evidencing good matching of the meander dipole antenna at
frequency $f_{2}$.

Figure 15: Measured magnitudes of reflection and transmission coefficients of
the implanted meander dipole antenna within a glass bottle filled with
distilled water; performed in the anechoic chamber.

At studied frequency $f_{2}$, measured magnitudes of reflection and
transmission coefficients are $|s_{11}|=-18\,\mathrm{dB}$ and
$|s_{12}|=-54\,\mathrm{dB}$, respectively. The transmission coefficient was
measured in the direction of gain pattern maximum $\theta=\pi/2$ and
$\varphi=0$. In all measurements, the vector network analyzer was calibrated
to the reference plane lying at coaxial ports placed at the end of a semi-
rigid cable feeding the antenna.

The gain pattern of the implanted meander dipole antenna,
$G(\theta,\varphi)=\eta_{\mathrm{rad}}D(\theta,\varphi)$, is obtained from the
measured transmission coefficient by excluding the gain of the reference
antenna and the path loss between the two antennas, which was calibrated as
$-36.10$ dB in total at frequency $f_{2}$ in the measurement setup. The
measured gain pattern was further corrected by the balun insertion loss of
0.30 dB and the cable loss of 0.35 dB (cable connected from the measurement
reference plane to the balun).

## References

  * [1] L. J. Chu, “Physical limitations of omni-directional antennas,” _J. Appl. Phys._ , vol. 19, pp. 1163–1175, Dec. 1948.
  * [2] H. A. Wheeler, “Fundamental limitations of small antennas,” _Proc. IRE_ , vol. 35, no. 12, pp. 1479–1484, 1947.
  * [3] R. M. Fano, “Theoretical limitations on the broadband matching of arbitrary impedances,” _Journal of the Franklin Inst._ , vol. 249, no. 1–2, pp. 57–83, 139–154, 1950.
  * [4] J. S. McLean, “A re-examination of the fundamental limits on the radiation Q of electrically small antennas,” _IEEE Trans. Antennas Propag._ , vol. 44, no. 5, pp. 672–675, May 1996.
  * [5] A. D. Yaghjian and S. R. Best, “Impedance, bandwidth and Q of antennas,” _IEEE Trans. Antennas Propag._ , vol. 53, no. 4, pp. 1298–1324, Apr. 2005\.
  * [6] R. F. Harrington, “On the gain and beamwidth of directional antennas,” _IRE Trans. Antennas Propag._ , vol. 6, no. 3, pp. 219–225, July 1958.
  * [7] M. Gustafsson and S. Nordebo, “Optimal antenna currents for Q, superdirectivity, and radiation patterns using convex optimization,” _IEEE Trans. Antennas Propag._ , vol. 61, no. 3, pp. 1109–1118, Mar. 2013\.
  * [8] S. Molesky, P. Chao, W. Jin, and A. W. Rodriguez, “Global T operator bounds on electromagnetic scattering: Upper bounds on far-field cross sections,” _Physical Review Research_ , vol. 2, no. 3, 2020.
  * [9] M. Gustafsson and M. Capek, “Maximum gain, effective area, and directivity,” _IEEE Trans. Antennas Propag._ , vol. 67, no. 8, pp. 5282 – 5293, Aug. 2019\.
  * [10] B. Gupta, S. Sankaralingam, and S. Dhar, “Development of wearable and implantable antennas in the last decade: A review,” in _2010 10th Mediterranean Microwave Symposium_. IEEE, 2010.
  * [11] D. Nikolayev, M. Zhadobov, L. L. Coq, P. Karban, and R. Sauleau, “Robust ultraminiature capsule antenna for ingestible and implantable applications,” _IEEE Trans. Antennas Propag._ , vol. 65, no. 11, pp. 6107–6119, Nov. 2017\.
  * [12] A. K. Skrivervik, M. Bosiljevac, and Z. Sipus, “Fundamental limits for implanted antennas: Maximum power density reaching free space,” _IEEE Trans. Antennas Propag._ , vol. 67, no. 8, pp. 4978 – 4988, Aug. 2019.
  * [13] M. Bosiljevac, Z. Sipus, and A. K. Skrivervik, “Propagation in finite lossy media: An application to WBAN,” _IEEE Antennas and Wireless Propagation Letters_ , vol. 14, pp. 1546–1549, 2015.
  * [14] M. Gao, D. Nikolayev, M. Bosiljevac, Z. Sipus, and A. K. Skrivervik, “Rules of thumb to assess losses of implanted antennas,” in _2021 15th European Conference on Antennas and Propagation (EuCAP)_. IEEE, 2021.
  * [15] M. Gao, Z. Sipus, and A. K. Skrivervik, “On the maximum power density of implanted antennas within simplified body phantoms,” in _2022 16th European Conference on Antennas and Propagation (EuCAP)_. IEEE, Mar. 2022.
  * [16] D. Nikolayev, W. Joseph, M. Zhadobov, R. Sauleau, and L. Martens, “Optimal radiation of body-implanted capsules,” _Physical Review Letters_ , vol. 122, no. 10, Mar. 2019.
  * [17] D. Nikolayev, M. Zhadobov, P. Karban, and R. Sauleau, “Electromagnetic radiation efficiency of body-implanted devices,” _Physical Review Applied_ , vol. 9, no. 2, Feb. 2018.
  * [18] L. Jelinek, J. Liska, M. Capek, V. Losenicky, and M. Gustafsson, “Maximum radiation efficiency of implanted antennas employing a novel hybrid method,” in _2021 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting (APS/URSI)_. IEEE, Dec. 2021.
  * [19] J. Liska, M. Gao, L. Jelinek, M. Capek, and A. Skrivervik, “Maximum radiation efficiency of an implantable antenna: The role of high-order modes,” in _2023 17th European Conference on Antennas and Propagation (EuCAP)_. IEEE, Feb. 2023.
  * [20] E. Algarp, “Antenna implants and feasibility of performance limitations: A study of radiation efficiency on electrically small antenna implants with finite conductivity and size,” Master’s thesis, KTH, School of Electrical Engineering and Computer Science (EECS), 2022.
  * [21] J. Liska, L. Jelinek, and M. Capek, “Fundamental bounds to time-harmonic quadratic metrics in electromagnetism: Overview and implementation,” _arXiv_ , 2021. [Online]. Available: https://arxiv.org/abs/2110.05312
  * [22] A. G. Polimeridis, M. T. H. Reid, S. G. Johnson, J. K. White, and A. W. Rodriguez, “On the computation of power in volume integral equation formulations,” _IEEE Trans. Antennas Propag._ , vol. 63, no. 2, pp. 611–620, Feb. 2015.
  * [23] J. Kim and Y. Rahmat-Samii, “Implanted antennas inside a human body: Simulations, designs, and characterizations,” _IEEE Transactions on Microwave Theory and Techniques_ , vol. 52, no. 8, pp. 1934–1943, Aug. 2004.
  * [24] V. Losenicky, L. Jelinek, M. Capek, and M. Gustafsson, “Method of moments and T-matrix hybrid,” _IEEE Trans. Antennas Propag._ , vol. 70, no. 5, pp. 3560–3574, 2022.
  * [25] J. Liska, L. Jelinek, and M. Capek, “Computation of fundamental bounds for antennas,” in _2022 16th European Conference on Antennas and Propagation (EuCAP)_. IEEE, Mar. 2022\.
  * [26] J. Nocedal and S. Wright, _Numerical Optimization_. New York, United States: Springer, 2006.
  * [27] (2022) CST Computer Simulation Technology. Dassault Systemes. [Online]. Available: https://www.3ds.com/products-services/simulia/products/cst-studio-suite/
  * [28] (2022) Altair FEKO. Altair. [Online]. Available: https://www.altair.com/feko
  * [29] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” _IEEE Trans. Antennas Propag._ , vol. 30, no. 3, pp. 409–418, May 1982.
  * [30] L. Morlein, L. Berkelmann, and D. Manteuffel, “Antenna de-embedding in FDTD using spherical wave functions by exploiting orthogonality,” 2021.
  * [31] L. Berkelmann, H. Jaschke, L. Morlein, L. Grundmann, and D. Manteuffel, “Antenna optimization for WBAN based on spherical wave functions de-embedding,” _IEEE Trans. Antennas Propag._ , pp. 1–1, 2022.
  * [32] R. F. Harrington, _Field Computation by Moment Methods_. Piscataway, New Jersey, United States: Wiley – IEEE Press, 1993.
  * [33] F. Merli, B. Fuchs, J. R. Mosig, and A. K. Skrivervik, “The effect of insulating layers on the performance of implanted antennas,” _IEEE Trans. Antennas Propag._ , vol. 59, no. 1, pp. 21–31, Jan. 2011.
  * [34] L. Jelinek and M. Capek, “Optimal currents on arbitrarily shaped surfaces,” _IEEE Trans. Antennas Propag._ , vol. 65, no. 1, pp. 329–341, Jan. 2017.
  * [35] D. M. Pozar, _Microwave Enginnering_ , 4th ed. Wiley, 2011.
  * [36] (2019) Antenna Toolbox for MATLAB (AToM). Czech Technical University in Prague. www.antennatoolbox.com. [Online]. Available: {www.antennatoolbox.com}

|  Jakub Liska received his B.Sc. and M.Sc. degrees in electrical engineering
from the Czech Technical University in Prague, Czech Republic, in 2019 and
2021, respectively. He is currently continuing towards a Ph.D. degree in
electrical engineering at the same university. His research interests include
EM field theory, fundamental bounds, computational electromagnetics, numerical
and convex optimization, numerical techniques, and eigenproblems.
---|---
|  Mingxiang Gao received the double B.Sc. degree in electrical engineering
and business administration and the M.Sc. degree in electrical engineering
from the Xi’an Jiaotong University, Xi’an, China, in 2016 and 2019,
respectively, and the M.Sc. degree (summa cum laude) in electrical engineering
from the Politecnico di Milano, Milan, Italy, in 2019. He is currently working
towards the Ph.D. degree in electrical engineering at the École Polytechnique
Fédérale de Lausanne, Lausanne, Switzerland. His current research interests
include the theory and design of implantable antennas, wireless techniques for
bioelectronics, and bioelectromagnetics.
---|---
|  Lukas Jelinek received his Ph.D. degree from the Czech Technical University
in Prague, Czech Republic, in 2006. In 2015 he was appointed Associate
Professor at the Department of Electromagnetic Field at the same university.
His research interests include wave propagation in complex media,
electromagnetic field theory, metamaterials, numerical techniques, and
optimization.
---|---
|  Erik R. Algarp received his M.Sc. degree in Electrical Engineering from KTH
Royal Institute of Technology, Sweden, in 2022. He is currently pursuing a
Ph.D degree in Electrical Engineering within the Microwaves and Antennas group
at École polytechnique fédérale de Lausanne.
---|---
|  Anja K. Skrivervik received the master’s and Ph.D. degrees in electrical
engineering from the École Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland, in 1986 and 1992, respectively. After a stay at the
University of Rennes, Rennes, France, as an Invited Research Fellow and two
years in the industry, she returned part-time to EPFL as an Assistant
Professor in 1996, where she is currently a Professeur Titulaire, the Head of
the Microwave and Antenna Group. She was the Director of the EE Section from
1996 to 2000, and she is currently the Director of the EE Doctoral School at
EPFL. She is also a Visiting Professor at the University of Lund. Her teaching
activities include courses on microwaves and antennas, and she teaches at
bachelor’s, master’s, and Ph.D. levels. She is the author or co-author of more
than 200 peer-reviewed scientific publications. Her research interests include
electrically small antennas, antennas in biological media, periodic
structures, reflect and transmit arrays, and numerical techniques for
electromagnetics. Dr. Skrivervik is a Board Member of the European School on
Antennas and is frequently requested to review research programs and centers
in Europe. She has been a member of the Board of Directors of the European
Association on Antennas and Propagation (EurAAP) since 2017. She received the
Latsis Award. She is very active in European collaboration and European
projects. She was the Chairperson of the Swiss URSI until 2012. She was the
General Chair of the Loughborough Antenna and Propagation Conference in 2015,
the Vice-Chair and Technical Program Committee-Chair of the EuCAP 2016
Conference, and the Financial Chair of EuCAP 2017 to EuCAP 2022.
---|---
|  Miloslav Capek (M’14, SM’17) received the M.Sc. degree in Electrical
Engineering 2009, the Ph.D. degree in 2014, and was appointed Associate
Professor in 2017, all from the Czech Technical University in Prague, Czech
Republic. He leads the development of the AToM (Antenna Toolbox for Matlab)
package. His research interests are in the area of electromagnetic theory,
electrically small antennas, antenna design, numerical techniques, and
optimization. He authored or co-authored over 130 journal and conference
papers. Dr. Capek is the Associate Editor of IET Microwaves, Antennas &
Propagation. He was a regional delegate of EurAAP between 2015 and 2020 and an
associate editor of Radioengineering between 2015 and 2018.
---|---
  *[EM]:
  *[MoM]:
  *[VSWE]: vector spherical wave expansion
  *[QCQP]: quadratically constrained quadratic program
  *[CST]: CST []
  *[FEKO]: FEKO []
  *[EFIE]:
  *[GEP]:
  *[AToM]: Antenna Toolbox for Matlab []