In this paper, we study a tumor growth model where the growth is driven by a diffusing nutrient and the tumor expands according to Darcy’s law with a mechanical pressure resulting from the incompressibility of the cells. Our focus is on the free boundary regularity of the tumor patch that holds beyond topological changes. A crucial element in our analysis is establishing the regularity of the hitting time T(x), namely the first time the tumor patch reaches a given point. We achieve this by introducing a novel Hamilton-Jacobi-Bellman (HJB) interpretation of the pressure, which is of independent interest. The HJB structure is obtained by viewing the model as a limit of the Porous Media Equation (PME) and building upon a new variant of the AB estimate. Using the HJB structure, we establish a new Hopf-Lax type formula for the pressure variable. Combined with barrier arguments, the formula allows us to show that T is <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2024_2081_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$C^{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mi>α</mi> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2024_2081_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha =\alpha (d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, which translates into a mild nondegeneracy of the tumor patch evolution. Building on this and obstacle problem theory, we show that the tumor patch boundary is regular in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2024_2081_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="85" /> </InlineMediaObject> <EquationSource Format="TEX">${ \mathbb {R}}^d\times (0,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo>×</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> except on a set of Hausdorff dimension at most <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2024_2081_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$d-\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>-</mo> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation>. On the set of regular points, we further show that the tumor patch is locally <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2024_2081_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">$C^{1,\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>α</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> in space-time. This conclusively establishes that instabilities in the boundary evolution do not amplify arbitrarily high frequencies.

Regularity and Nondegeneracy for Tumor Growth with Nutrients

期刊论文
发表日期：

2025年1月11日

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Carson Collins,
Matt Jacobs,
Inwon Kim

摘要

In this paper, we study a tumor growth model where the growth is driven by a diffusing nutrient and the tumor expands according to Darcy’s law with a mechanical pressure resulting from the incompressibility of the cells. Our focus is on the free boundary regularity of the tumor patch that holds beyond topological changes. A crucial element in our analysis is establishing the regularity of the hitting time T(x), namely the first time the tumor patch reaches a given point. We achieve this by introducing a novel Hamilton-Jacobi-Bellman (HJB) interpretation of the pressure, which is of independent interest. The HJB structure is obtained by viewing the model as a limit of the Porous Media Equation (PME) and building upon a new variant of the AB estimate. Using the HJB structure, we establish a new Hopf-Lax type formula for the pressure variable. Combined with barrier arguments, the formula allows us to show that T is $C^{\alpha }$ $C^{α}$ with $\alpha =\alpha (d)$ $α = α (d)$ , which translates into a mild nondegeneracy of the tumor patch evolution. Building on this and obstacle problem theory, we show that the tumor patch boundary is regular in ${ \mathbb {R}}^d\times (0,\infty )$ $R^{d} \times (0, \infty)$ except on a set of Hausdorff dimension at most $d-\alpha $ $d - α$ . On the set of regular points, we further show that the tumor patch is locally $C^{1,\alpha }$ $C^{1, α}$ in space-time. This conclusively establishes that instabilities in the boundary evolution do not amplify arbitrarily high frequencies.